GermanWikipedia - /page/entry.mu

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<title>Adelering</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Adelering</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Der Adelering wird in der <a href="Zahlentheorie" title="Zahlentheorie">Zahlentheorie</a>, einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>, definiert. Er steht im Zusammenhang mit der <a href="Klassenk%C3%B6rpertheorie" title="Klassenkörpertheorie">Klassenkörpertheorie</a>. Der Adelering ist das <a href="Eingeschr%C3%A4nktes_direktes_Produkt" title="Eingeschränktes direktes Produkt">restringierte Produkt</a> aller Vervollständigungen eines <a href="Globaler_K%C3%B6rper" title="Globaler Körper">globalen Körpers</a>. Damit enthält er alle diese Vervollständigungen.

Der Adelering ist ein <a href="Charakter_(Mathematik)" title="Charakter (Mathematik)">selbstdualer</a>, <a href="Topologischer_Ring" title="Topologischer Ring">topologischer Ring</a>, welcher auf Grundlage eines globalen Körpers konstruiert wird. Er ermöglicht eine besonders elegante Darstellung des <a href="Artinsches_Reziprozit%C3%A4tsgesetz" title="Artinsches Reziprozitätsgesetz">Artinschen Reziprozitätsgesetzes</a>.

Die <a href="Idelklassengruppe" class="mw-redirect" title="Idelklassengruppe">Idelklassengruppe</a>, welche der Quotient aus den <a href="Einheit_(Mathematik)" title="Einheit (Mathematik)">Einheiten</a> des Adelerings und den Einheiten des Körpers ist, stellt ein zentrales Objekt in der Klassenkörpertheorie dar.

Notation: Im Folgenden ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein <a href="Globaler_K%C3%B6rper" title="Globaler Körper">globaler Körper</a>. Das bedeutet, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> entweder ein <a href="Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper">algebraischer Zahlkörper</a> oder ein <a href="Algebraischer_Funktionenk%C3%B6rper" class="mw-redirect" title="Algebraischer Funktionenkörper">algebraischer Funktionenkörper</a> positiver <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> vom <a href="Transzendenzgrad" class="mw-redirect" title="Transzendenzgrad">Transzendenzgrad</a> 1 ist. Im ersten Fall bedeutet das, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce6782a2935122dafec2947d1d7ba168be4399ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }" loading="lazy"> eine <a href="Endliche_K%C3%B6rpererweiterung" class="mw-redirect" title="Endliche Körpererweiterung">endliche Körpererweiterung</a> ist, im zweiten Fall, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {F} _{p^{r}}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle K/\mathbb {F} _{p^{r}}(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6990c9b17ec373b4f2dae2fe1663ef3133c0cfea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.123ex; height:3.009ex;" alt="{\displaystyle K/\mathbb {F} _{p^{r}}(t)}" loading="lazy"> eine endliche Körpererweiterung ist.

Im Folgenden bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> eine <a href="Bewertungstheorie#Definition_2" class="mw-redirect" title="Bewertungstheorie">Stelle</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

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<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Die triviale <a href="Bewertungstheorie" class="mw-redirect" title="Bewertungstheorie">Bewertung</a> und der dazu korrespondierende triviale Betrag werden im kompletten Artikel ausgeschlossen. Es wird unterschieden zwischen endlichen (<a href="Nicht-archimedische_Bewertung" class="mw-redirect" title="Nicht-archimedische Bewertung">nicht-archimedischen</a>) <a href="Bewertungstheorie#Allgemeine_Bewertungen" class="mw-redirect" title="Bewertungstheorie">Stellen</a>, welche als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v<\infty }">

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<annotation encoding="application/x-tex">{\displaystyle v<\infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/430e8a29fad3bde77f47b8a3075126109617ee44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.55ex; height:1.843ex;" alt="{\displaystyle v<\infty }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\nmid \infty }">

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<annotation encoding="application/x-tex">{\displaystyle v\nmid \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef1f7d6fb89482044b4c147ca02ebdf6d891115b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.389ex; height:2.843ex;" alt="{\displaystyle v\nmid \infty }" loading="lazy"> notiert werden, und unendlichen (archimedischen) Stellen, welche als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\mid \infty }">

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<annotation encoding="application/x-tex">{\displaystyle v\mid \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d77dd4c6fc35e8c9a32076d6ebe997745a8aa6ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.389ex; height:2.843ex;" alt="{\displaystyle v\mid \infty }" loading="lazy"> notiert werden.

Im Folgenden bezeichne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> die <a href="Endliche_Menge" title="Endliche Menge">endliche Menge</a> der unendlichen Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

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<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Wir schreiben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> für eine endliche Teilmenge der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

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<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> welche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> enthält. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> die Vervollständigung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> nach einer Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v.}">

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<annotation encoding="application/x-tex">{\displaystyle v.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb7b87f7afb1452b9d0e287f8ef746a9912c8333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle v.}" loading="lazy"> Bei einer <a href="Diskrete_Bewertung" class="mw-redirect" title="Diskrete Bewertung">diskreten Bewertung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> bezeichne mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4714e4b7795af9e2c3d727be09bd091a4d1076.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{v}}" loading="lazy"> den zugehörigen diskreten Bewertungsring von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7243d708afcda5b7486f6a276fc555622170baeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.812ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {m}}_{v}}" loading="lazy"> das <a href="Maximales_Ideal" title="Maximales Ideal">maximale Ideal</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d6f594d1b3f1caf191e2bc9842685a7e686ed34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{v}.}" loading="lazy"> Ist dieses ein <a href="Hauptideal" title="Hauptideal">Hauptideal</a>, so schreibe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{v}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f21165a2b8dd291d122fe5b30a9625e978f671e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.355ex; height:2.009ex;" alt="{\displaystyle \pi _{v}}" loading="lazy"> für ein uniformisierendes Element. Der Leser sei weiterhin auf die eineindeutige Identifikation von Beträgen und Bewertungen eines Körpers hingewiesen bei Fixierung einer geeigneten Konstante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C>1:}">

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<annotation encoding="application/x-tex">{\displaystyle C>1:}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de5960bdab30dbe15a45a16c4605014e10dc69d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.319ex; height:2.176ex;" alt="{\displaystyle C>1:}" loading="lazy">

Die Bewertung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> wird dem Betrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f33761816b48a7198cfcc0109ba13185adb5ba9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.002ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{v}}" loading="lazy"> zugeordnet, welcher wie folgt definiert wird:

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<mtext> falls </mtext>

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<mtext> falls </mtext>

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<annotation encoding="application/x-tex">{\displaystyle |x|_{v}:={\begin{cases}C^{-v(x)}&,{\text{ falls }}x\neq 0\\0&,{\text{ falls }}x=0\end{cases}}\quad \forall x\in K.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e491c4a6962bba3cdb9668ae89ffad25a906622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.909ex; height:6.176ex;" alt="{\displaystyle |x|_{v}:={\begin{cases}C^{-v(x)}&,{\text{ falls }}x\neq 0\\0&,{\text{ falls }}x=0\end{cases}}\quad \forall x\in K.}" loading="lazy"></dd></dl>

Umgekehrt wird dem Betrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |}">

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<annotation encoding="application/x-tex">{\displaystyle |\cdot |}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4570d0a1c9fb8f2f413f0b73ce846dd1eb1dca3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.973ex; height:2.843ex;" alt="{\displaystyle |\cdot |}" loading="lazy"> die Bewertung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{|\cdot |}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{|\cdot |}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db1e2a591a869a5a769af43038a4b46bfac7534d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle v_{|\cdot |}}" loading="lazy"> zugeordnet, welche wie folgt definiert ist: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{|\cdot |}(x):=-\log _{C}(|x|)}">

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<annotation encoding="application/x-tex">{\displaystyle v_{|\cdot |}(x):=-\log _{C}(|x|)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b70868df87d09d70abf994712a79dc62083c5229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.697ex; height:3.176ex;" alt="{\displaystyle v_{|\cdot |}(x):=-\log _{C}(|x|)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K^{\times }.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in K^{\times }.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88f6a80ff88716f86a46dda1f80e1084f182cb0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.422ex; height:2.343ex;" alt="{\displaystyle x\in K^{\times }.}" loading="lazy"> Diese Identifikation wird im Artikel laufend verwendet.

Im Artikel wird das restringierte Produkt mit

<msup>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\prod }}^{{\mathcal {O}}_{v}}K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/418b3e794ad846564ed2646a4078ea5628f74eff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.324ex; height:5.009ex;" alt="{\displaystyle {\widehat {\prod }}^{{\mathcal {O}}_{v}}K_{v}}" loading="lazy">

notiert. Eine andere geläufige Notation dafür ist

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod \limits (K_{v},{\mathcal {O}}_{v}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0aa6d26e7633068ccd24670d02e7228458dc4abd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.342ex; height:3.843ex;" alt="{\displaystyle \prod \limits (K_{v},{\mathcal {O}}_{v}).}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Begriffserklärung">Begriffserklärung</h2></div>

In der lokalen Klassenkörpertheorie spielt die <a href="Multiplikative_Gruppe" class="mw-redirect" title="Multiplikative Gruppe">multiplikative Gruppe</a> des lokalen <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körpers</a> eine wichtige Rolle. In der globalen Klassenkörpertheorie wird diese Rolle von der <a href="Idelklassengruppe" class="mw-redirect" title="Idelklassengruppe">Idelklassengruppe</a> übernommen. Der Begriff des Idels ist eine Abänderung des Idealbegriffs, wobei beide Begriffe in Beziehung zueinander stehen, siehe dazu den Satz über den <a href="Idelegruppe#Zusammenhang_zwischen_Idealklassengruppe_und_Idelklassengruppe" title="Idelegruppe">Zusammenhang zwischen der Ideal- und der Idelklassengruppe</a>. Der Idelbegriff wurde von dem französischen Mathematiker <a href="Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a> (1909–1984) unter dem Namen „ideal element“ (abgekürzt: id.el.) eingeführt. Der Begriff des Adels geht zurück auf die ursprüngliche Bezeichnung „additives Idel“. Bei der Aussprache von Adel liegt die Betonung auf der 2. Silbe.

Die Idee hinter dem Adelering ist es, dass man alle <a href="Vervollst%C3%A4ndigung_(Kommutative_Algebra)" title="Vervollständigung (Kommutative Algebra)">Vervollständigungen</a> des globalen Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> auf einmal betrachtet. Auf den ersten Blick scheint die Definition über das kartesische Produkt sinnvoll, jedoch wird der Adelering mit dem restringierten Produkt definiert, wie im nächsten Abschnitt erläutert wird. Dies hat mehrere Gründe:

<ul><li>Wenn man den globalen Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> in das Produkt über die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K_{v}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K_{v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd650f9f1ada1e86b814e2a10eb50b6d38f8455e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle {K_{v}}}" loading="lazy"> einbettet, dann gilt für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {k}\in {K}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {k}\in {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99f40b6f51170506379359a800b879d365644be7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.118ex; height:2.176ex;" alt="{\displaystyle {k}\in {K}}" loading="lazy">: für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {v}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96ee8864003f6d1ab82b4a53941021863c19b74a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle {v}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(k)=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(k)=0,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7863c2db98fb86ee2a880eaae7ddc9d64ebbafa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.056ex; height:2.843ex;" alt="{\displaystyle v(k)=0,}" loading="lazy"> also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {|k|_{v}}=1}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {|k|_{v}}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a269ae402e7a45aea3232b55a8627987b056f419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.795ex; height:3.009ex;" alt="{\displaystyle {|k|_{v}}=1}" loading="lazy"> (vgl. <a href="Globaler_K%C3%B6rper" title="Globaler Körper">globaler Körper</a>). Die Terminologie „fast alle“ meint im gesamten Artikel immer „alle bis auf endlich viele“. Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> sogar in das restringierte Produkt einbettbar.</li>

<li>Der Adelering wird dadurch zu einem lokalkompakten, topologischen Ring. Das unrestringierte Produkt hingegen ist nicht lokalkompakt. Daher ist auf dem unrestringierten Produkt keine <a href="Harmonische_Analyse" title="Harmonische Analyse">Harmonische Analyse</a> möglich.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Definition_des_Adelerings_eines_globalen_Körpers_K">Definition des Adelerings eines globalen Körpers K</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Die_Menge_der_endlichen_Adele_eines_globalen_Körpers_K">Die Menge der endlichen Adele eines globalen Körpers K</h3></div>

Die Menge der endlichen Adele eines globalen Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12aa23fa0f2bf98611ab7125fc1c064e2461f8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle {K},}" loading="lazy"> geschrieben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K,fin},}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K,fin},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac1e2748064e878939bc0b2c69cc075d435a502b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.933ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K,fin},}" loading="lazy"> ist definiert als das restringierte Produkt der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K_{v}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K_{v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd650f9f1ada1e86b814e2a10eb50b6d38f8455e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle {K_{v}}}" loading="lazy"> mit Restriktionsbedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\mathcal {O}}_{v}},}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {{\mathcal {O}}_{v}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9064974e4c308ec97dd67547e7665107ad64a3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle {{\mathcal {O}}_{v}},}" loading="lazy"> das heißt

<msub>

</mrow>

</mrow>

</msub>

<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K,fin}:={\widehat {\prod \limits _{v\nmid \infty }}}^{{\mathcal {O}}_{v}}K_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e9944c91a128fb487788266ef5be5be83be83d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:19.002ex; height:7.176ex;" alt="{\displaystyle \mathbb {A} _{K,fin}:={\widehat {\prod \limits _{v\nmid \infty }}}^{{\mathcal {O}}_{v}}K_{v}.}" loading="lazy"></dd></dl>

Das bedeutet, dass die Menge der endlichen Adele alle Elemente der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle (x_{v})_{v}\in \prod _{v\nmid \infty }K_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle (x_{v})_{v}\in \prod _{v\nmid \infty }K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e3ab6507450fcfda0cd83f1f1d2d2630e54e35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.752ex; height:3.343ex;" alt="{\displaystyle \textstyle (x_{v})_{v}\in \prod _{v\nmid \infty }K_{v}}" loading="lazy"> enthält, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{v}\in {\mathcal {O}}_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{v}\in {\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5d72637f8f96a46865977c20ad0b241c46bb91e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.079ex; height:2.509ex;" alt="{\displaystyle x_{v}\in {\mathcal {O}}_{v}}" loading="lazy"> für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb7b87f7afb1452b9d0e287f8ef746a9912c8333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle v.}" loading="lazy"> Die Addition und Multiplikation werden komponentenweise erklärt. Dadurch wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K,fin}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add86865a65b773dd2a2838e84e89763b5013a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.286ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K,fin}}" loading="lazy"> zu einem Ring. Wir installieren auf der Menge der endlichen Adele die restringierte Produkttopologie. Das ist diejenige Topologie, die von den sogenannten restringierten offenen Rechtecken erzeugt wird, welche folgende Form haben:

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=\prod _{v\in E}U_{v}\times \prod _{v\notin E}{\mathcal {O}}_{v},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0bb7061542d75474cb9a2085a94ee73001a8d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.937ex; height:6.009ex;" alt="{\displaystyle U=\prod _{v\in E}U_{v}\times \prod _{v\notin E}{\mathcal {O}}_{v},}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> eine endliche Teilmenge der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist, welche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> enthält und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{v}\subset K_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{v}\subset K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c0d22d4c350a7a725fa289a17efb9e39cb143f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.718ex; height:2.509ex;" alt="{\displaystyle U_{v}\subset K_{v}}" loading="lazy"> offen sind.

Bemerkung: In der deutschen Literatur wird auch der Name <a href="Eingeschr%C3%A4nktes_direktes_Produkt" title="Eingeschränktes direktes Produkt">eingeschränktes direktes Produkt</a> für das restringierte Produkt verwendet. Im Folgenden wird der Begriff restringiertes Produkt verwendet. Weiterhin wird im Folgenden endlicher Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> als Synonym für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K,fin}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add86865a65b773dd2a2838e84e89763b5013a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.286ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K,fin}}" loading="lazy"> verwendet.

<div class="mw-heading mw-heading3"><h3 id="Der_Adelering_eines_globalen_Körpers_K">Der Adelering eines globalen Körpers K</h3></div>

Der Adelering des globalen Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12aa23fa0f2bf98611ab7125fc1c064e2461f8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle {K},}" loading="lazy"> geschrieben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K},}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bef6ca8bc0051c81dee17fa2d17e74c6029d22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K},}" loading="lazy"> ist definiert als das Produkt der Menge der endlichen Adele mit dem Produkt der endlich vielen Vervollständigungen nach den unendlichen Stellen. Diese sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> und treten nur im algebraischen Zahlkörperfall auf. Damit erhalten wir also:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}:=\mathbb {A} _{K,fin}\times \prod \limits _{v\mid \infty }K_{v}={\widehat {\prod \limits _{v\nmid \infty }}}^{{\mathcal {O}}_{v}}K_{v}\times \prod \limits _{v\mid \infty }K_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43a57b8e8eb5c86fc0d1a5d2bd2e1262ad8d76c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:43.871ex; height:7.176ex;" alt="{\displaystyle \mathbb {A} _{K}:=\mathbb {A} _{K,fin}\times \prod \limits _{v\mid \infty }K_{v}={\widehat {\prod \limits _{v\nmid \infty }}}^{{\mathcal {O}}_{v}}K_{v}\times \prod \limits _{v\mid \infty }K_{v}.}" loading="lazy"></dd></dl>

In Fall eines Funktionenkörpers ist die Menge der endlichen Adele gleich dem Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27fe4c62b2f1eb66509fb022e845b6d5a6e38019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle {K}.}" loading="lazy"> Auf dem Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> wird eine Addition und Multiplikation jeweils komponentenweise erklärt. Dadurch wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> zu einem Ring. Die Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> werden die Adele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> genannt. Wir schreiben im Folgenden den Adelering als

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}={\widehat {\prod _{v}}}K_{v},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43e6c58dac87bc3e9b011474ca54793fee8de81a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.089ex; height:6.176ex;" alt="{\displaystyle \mathbb {A} _{K}={\widehat {\prod _{v}}}K_{v},}" loading="lazy"></dd></dl>

obwohl dies kein restringiertes Produkt im eigentlichen Sinne ist. Im Folgenden wird nicht extra darauf hingewiesen, dass die unendlichen Stellen unrestringiert dem Produkt hinzugefügt werden.

<div class="mw-heading mw-heading3"><h3 id="Die_Menge_der_S-Adele_eines_globalen_Körpers_K">Die Menge der S-Adele eines globalen Körpers K</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> ein globaler Körper und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {S}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {S}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfd749ff84450814edb04bb695d674c8dd6a2d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle {S}}" loading="lazy"> eine Teilmenge der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27fe4c62b2f1eb66509fb022e845b6d5a6e38019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle {K}.}" loading="lazy"> Definiere die Menge der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">-Adele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> als

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {A} _{K,S}:={\widehat {\prod \limits _{v\in S}}}^{{\mathcal {O}}_{v}}K_{v}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/102aec67823e01ef1790fa0a190114452de5eb66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.602ex; margin-bottom: -0.236ex; width:18.356ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\mathbb {A} _{K,S}:={\widehat {\prod \limits _{v\in S}}}^{{\mathcal {O}}_{v}}K_{v}.\end{aligned}}}" loading="lazy"></dd></dl>

Die unendlichen Stellen, sofern in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {S}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {S}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfd749ff84450814edb04bb695d674c8dd6a2d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle {S}}" loading="lazy"> enthalten, werden dabei ohne Restriktionsbedingung hinzugefügt. Definiere weiterhin

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}^{S}:={\widehat {\prod \limits _{v\notin S}}}^{{\mathcal {O}}_{v}}K_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a95d6209885b8c6ef2c5913128df5f76fbbb04de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.087ex; height:7.176ex;" alt="{\displaystyle \mathbb {A} _{K}^{S}:={\widehat {\prod \limits _{v\notin S}}}^{{\mathcal {O}}_{v}}K_{v}.}" loading="lazy"></dd></dl>

Es gilt dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1d3daa284ae984d9e96976457ae342f86a9889e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.217ex; height:3.343ex;" alt="{\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Der_rationale_Adelering_𝔸ℚ">Der rationale Adelering 𝔸ℚ</h3></div>

Wir betrachten den Spezialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b65039e866ecde3cb208fa49082994d3b17f938b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.619ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} .}" loading="lazy"> Zuerst überlegen wir uns, wie die Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> aussieht: Der <a href="Satz_von_Ostrowski" title="Satz von Ostrowski">Satz von Ostrowski</a> besagt, dass die Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p\in \mathbb {N} :p{\text{ prim }}\}\cup \{\infty \}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{p\in \mathbb {N} :p{\text{ prim }}\}\cup \{\infty \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71b80a5a809024eae58f652784dfe68715ad3219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.299ex; height:2.843ex;" alt="{\displaystyle \{p\in \mathbb {N} :p{\text{ prim }}\}\cup \{\infty \}}" loading="lazy"> identifiziert werden kann, wobei die Primzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> dabei die Äquivalenzklasse des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-adischen Betrag repräsentiert und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"> für die folgende Äquivalenzklasse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\infty }}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cdfb936407ce40d20152b514afb6f4eb5a055e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.848ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{\infty }}" loading="lazy"> steht, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\infty }}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cdfb936407ce40d20152b514afb6f4eb5a055e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.848ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{\infty }}" loading="lazy"> wie folgt definiert wird:

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> falls </mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> sonst </mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x|_{\infty }:={\begin{cases}x&,{\text{ falls }}x\geq 0\\-x&,{\text{ sonst }}\end{cases}}\quad \forall x\in \mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c7371f70e304e3d049be44d979d315a47ebfaf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.31ex; height:6.176ex;" alt="{\displaystyle |x|_{\infty }:={\begin{cases}x&,{\text{ falls }}x\geq 0\\-x&,{\text{ sonst }}\end{cases}}\quad \forall x\in \mathbb {Q} .}" loading="lazy"></dd></dl>

Als Nächstes stellen wir fest, dass die Vervollständigung nach den Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> gerade die Körper der p-adischen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35f44bc6894c682710705f3ea74f33042e0acc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" loading="lazy"> für eine Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> bzw. der Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> für die Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"> sind. Der zugehörige Ganzzahlring zum Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35f44bc6894c682710705f3ea74f33042e0acc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2d1fd592239d60f9a9f36feb4b3be40df666f6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}.}" loading="lazy"> Damit folgt, dass der endliche Adelering der rationalen Zahlen gleich

<msub>

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</mrow>

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<msup>

<mover>

</mrow>

</munder>

</mover>

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<msub>

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</mrow>

</msup>

<msub>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}={\widehat {\prod \limits _{p<\infty }}}^{\mathbb {Z} _{p}}\mathbb {Q} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ef0ab080464eaa2a44f461b7af6db6d036d1c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.982ex; height:6.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}={\widehat {\prod \limits _{p<\infty }}}^{\mathbb {Z} _{p}}\mathbb {Q} _{p}}" loading="lazy"></dd></dl>

ist. Der ganze Adelering ist damit gleich

<msub>

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</mrow>

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<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

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</msup>

<msub>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }=({\widehat {\prod \limits _{p<\infty }}}^{\mathbb {Z} _{p}}\mathbb {Q} _{p})\times \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebcc100f6d2051ccc420fb5910e25d2571250c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.042ex; height:6.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }=({\widehat {\prod \limits _{p<\infty }}}^{\mathbb {Z} _{p}}\mathbb {Q} _{p})\times \mathbb {R} ,}" loading="lazy"></dd></dl>

wofür wir auch verkürzt schreiben:

<msub>

</mrow>

</mrow>

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<mover>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }={\widehat {\prod \limits _{p\leq \infty }}}\mathbb {Q} _{p},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f61aee6a960dc2eba93a5dbb61dde62664fd04a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.55ex; height:6.509ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }={\widehat {\prod \limits _{p\leq \infty }}}\mathbb {Q} _{p},}" loading="lazy"></dd></dl>

mit der Konvention <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{\infty }:=\mathbb {R} .}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{\infty }:=\mathbb {R} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6d0d5b1d0014924a226a8ad9f870a2d642de6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.754ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} _{\infty }:=\mathbb {R} .}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Unterschied_zwischen_restringierter_und_unrestringierter_Produkttopologie">Unterschied zwischen restringierter und unrestringierter Produkttopologie</h3></div>

Die Folge von Adelen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }}">

<msub>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af9d3468eef6e7ce070ec0c3125ba4bf03ad145d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.189ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }}" loading="lazy">

<mtr>

<mtd>

<msub>

</mrow>

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</mtd>

<mtd>

<mrow>

<mrow>

<mfrac>

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<mtr>

<mtd>

<msub>

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<mtd>

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<mrow>

<mfrac>

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</mtr>

<mtr>

<mtd>

<msub>

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</msub>

</mtd>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

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</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}&=\left({\frac {1}{2}},1,1,\dotsc \right)\\x_{2}&=\left(1,{\frac {1}{3}},1,\dotsc \right)\\x_{3}&=\left(1,1,{\frac {1}{5}},1,\dotsc \right)\\x_{4}&=\left(1,1,1,{\frac {1}{7}},1,\dotsc \right)\\&\vdots \end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b16319e6b53359890cb843dc30a32f81ba85f678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.005ex; width:24.196ex; height:29.176ex;" alt="{\displaystyle {\begin{aligned}x_{1}&=\left({\frac {1}{2}},1,1,\dotsc \right)\\x_{2}&=\left(1,{\frac {1}{3}},1,\dotsc \right)\\x_{3}&=\left(1,1,{\frac {1}{5}},1,\dotsc \right)\\x_{4}&=\left(1,1,1,{\frac {1}{7}},1,\dotsc \right)\\&\vdots \end{aligned}}}" loading="lazy"></dd></dl>

konvergiert in der Produkttopologie gegen das Einsadel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(1,1,\dotsc ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=(1,1,\dotsc ),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20d8e10fa4dcf01a82997984a697d85a4bae7178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14ex; height:2.843ex;" alt="{\displaystyle x=(1,1,\dotsc ),}" loading="lazy"> jedoch nicht in der restringierten Produkttopologie.

Beweis: Die Konvergenz in der Produkttopologie entspricht der koordinatenweisen Konvergenz. Diese ist trivial, da die Koordinatenfolgen stationär werden. Die Folge konvergiert nicht in der restringierten Produkttopologie, da für jedes Adel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=(a_{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a674f92e164aa610583d79b4b6ec0e64f5d084d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.515ex; height:3.009ex;" alt="{\displaystyle a=(a_{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}" loading="lazy"> und für jedes restringierte offene Rechteck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle U=\prod _{p\in E}U_{p}\times \prod _{p\notin E}\mathbb {Z} _{p}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle U=\prod _{p\in E}U_{p}\times \prod _{p\notin E}\mathbb {Z} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51c72ab5eb1c7379b17b728e568c74f660b4d424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.961ex; height:3.343ex;" alt="{\displaystyle \textstyle U=\prod _{p\in E}U_{p}\times \prod _{p\notin E}\mathbb {Z} _{p}}" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}">

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bb1f2dc2902570a4e0622ae6e93d732abba11d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.243ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{p}\in \mathbb {Z} _{p}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{p}\in \mathbb {Z} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69138e20b01dc5af0a53b74fb56a486a0af89e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.739ex; height:2.843ex;" alt="{\displaystyle a_{p}\in \mathbb {Z} _{p}}" loading="lazy"> und daher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}">

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bb1f2dc2902570a4e0622ae6e93d732abba11d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.243ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\notin F.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\notin F.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5071d98b459e5e611a0ca5b1660c86fbc410d5ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.487ex; height:2.676ex;" alt="{\displaystyle p\notin F.}" loading="lazy"> Es folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n}-a)\notin U}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n}-a)\notin U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e152c9483179c1d6fc483383814d92efa83ef97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.051ex; height:2.843ex;" alt="{\displaystyle (x_{n}-a)\notin U}" loading="lazy"> für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6b82f4024a32a795b10f00971e03ad88ebaa367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}" loading="lazy"> Hierbei stehen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> für endliche Teilmengen der Stellenmenge. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> eine endliche Ausnahmemenge des Adels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b803da9c45c1186883bde55107e9ccb102c92c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}" loading="lazy">

Der Adelering trägt nicht die Teilraumtopologie der Produkttopologie, da ansonsten der Adelering keine lokalkompakte Gruppe ist, vgl. hierzu den <a href="#Der_Adelering_ist_ein_lokalkompakter,_topologischer_Ring">Satz, dass der Adelering ein topologischer Ring ist</a>.

<div class="mw-heading mw-heading3"><h3 id="Diagonale_Einbettung_des_globalen_Körpers_in_seinen_Adelering">Diagonale Einbettung des globalen Körpers in seinen Adelering</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> ein globaler Körper. Es gibt eine natürliche diagonale Einbettung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> in seinen Adelering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}:}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}:}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe77ca8e0aeb649f3475cd7114ad71f0871d6ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.663ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}:}" loading="lazy">

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&K\hookrightarrow \mathbb {A} _{K},\\&a\mapsto (a,a,a,\dotsc ).\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ad6fa94322dfc79f13e949a36b0305f5d7a8b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.566ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}&K\hookrightarrow \mathbb {A} _{K},\\&a\mapsto (a,a,a,\dotsc ).\end{aligned}}}" loading="lazy"></dd></dl>

Die Einbettung ist wohldefiniert, da für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a3581f6410248bed60440badc5bd362ca1cc8c82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.394ex; height:2.176ex;" alt="{\displaystyle \alpha \in K}" loading="lazy"> gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in {\mathcal {O}}_{v}^{\times }}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in {\mathcal {O}}_{v}^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3c3be9a46736ba547005b25d0e198475a59cc7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.689ex; height:2.843ex;" alt="{\displaystyle \alpha \in {\mathcal {O}}_{v}^{\times }}" loading="lazy"> für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {v}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1e60e441c75d369888a5c10d6b675e4d0e5c11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle {v}.}" loading="lazy"> Sie ist injektiv, denn die Einbettung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K_{v}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K_{v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd650f9f1ada1e86b814e2a10eb50b6d38f8455e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle {K_{v}}}" loading="lazy"> ist bereits injektiv für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {v}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1e60e441c75d369888a5c10d6b675e4d0e5c11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle {v}.}" loading="lazy"> Es folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> als Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> aufgefasst werden kann. Man kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58e8935f16888d2f0621f2c4f3021469dcef9e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle {K}}" loading="lazy"> sogar als Unterring seines Adelerings auffassen. Die Elemente aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\subset \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/035637aa0de3a7578f8fcc6a8126a8237162455e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.536ex; height:2.509ex;" alt="{\displaystyle K\subset \mathbb {A} _{K}}" loading="lazy"> werden die Hauptadele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> genannt.

<div class="mw-heading mw-heading3"><h3 id="Die_Idelegruppe">Die Idelegruppe</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Idelegruppe" title="Idelegruppe">Idelegruppe</a></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein <a href="Globaler_K%C3%B6rper" title="Globaler Körper">globaler Körper</a>. Die Einheitengruppe des Adelerings

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{K}:=\mathbb {A} _{K}^{\times },}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88830e2ec9917a47ae8b4c3edf0cc0b52dbe093d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.48ex; height:2.843ex;" alt="{\displaystyle I_{K}:=\mathbb {A} _{K}^{\times },}" loading="lazy"></dd></dl>

mit der mittels der Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto (x,x^{-1})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto (x,x^{-1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/940fd11f605707c3b32227ae571cfba4036b0fcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.779ex; height:3.176ex;" alt="{\displaystyle x\mapsto (x,x^{-1})}" loading="lazy"> durch die <a href="Produkttopologie" title="Produkttopologie">Produkttopologie</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}^{\times }\times \mathbb {A} _{K}^{\times }}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}^{\times }\times \mathbb {A} _{K}^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a651b68fc7bc2e6b3dd1e0663f161af614bf0a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.583ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}^{\times }\times \mathbb {A} _{K}^{\times }}" loading="lazy"> erzeugten <a href="Teilraumtopologie" title="Teilraumtopologie">Teilraumtopologie</a>, ist die sogenannte Idelegruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Alternative_Definition_des_Adelerings_eines_Zahlkörpers">Alternative Definition des Adelerings eines Zahlkörpers</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Die_pro-endliche_Komplettierung_der_ganzen_Zahlen">Die pro-endliche Komplettierung der ganzen Zahlen</h3></div>

Definiere

<mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }}:=\lim \limits _{\overleftarrow {n}}\mathbb {Z} /n\mathbb {Z} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac227d12a14516a77c4704af2d4df6a8f6d8aeec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.217ex; height:5.509ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }}:=\lim \limits _{\overleftarrow {n}}\mathbb {Z} /n\mathbb {Z} ,}" loading="lazy"></dd></dl>

d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbb {Z} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12112fb1afd12580b51b080bb33356edd1d0c797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }}}" loading="lazy"> ist die <a href="Proendliche_Vervollst%C3%A4ndigung" title="Proendliche Vervollständigung">pro-endliche Komplettierung</a> der Ringe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2120ebbc85f91df66c6de5446367bf9fd620844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }" loading="lazy"> mit der partiellen Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq m:\Leftrightarrow m\mid n.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq m:\Leftrightarrow m\mid n.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1feab6283f293b8150bac959432bfcfcc7d69499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.814ex; height:2.843ex;" alt="{\displaystyle n\geq m:\Leftrightarrow m\mid n.}" loading="lazy"> Die pro-endliche Komplettierung der ganzen Zahlen ist also der <a href="Limes_(Kategorientheorie)" title="Limes (Kategorientheorie)">projektive Limes</a> über die Ringe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} .}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e28a230b549d86a8e7b93ae1383579ddc7178d88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} .}" loading="lazy">

Mit Hilfe des chinesischen Restsatzes kann gezeigt werden, dass die pro-endliche Komplettierung der ganzen Zahlen isomorph zum Produkt der ganzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-adischen Zahlen ist. Es gilt also

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }}\cong \prod \limits _{p{\text{ prim }}}\mathbb {Z} _{p}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a3e8bda46bcf04c5b9a649afade80775d822bb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.665ex; height:5.843ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }}\cong \prod \limits _{p{\text{ prim }}}\mathbb {Z} _{p}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Alternative_Definition_des_Adelerings_eines_Zahlkörpers_2">Alternative Definition des Adelerings eines Zahlkörpers</h3></div>

Definiere nun den Ring (der ganzzahligen Adele)

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Z} }:={\widehat {\mathbb {Z} }}\times \mathbb {R} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6451e320e74dd3a1e2a3d9a5bdcd974eada9287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.468ex; height:3.176ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Z} }:={\widehat {\mathbb {Z} }}\times \mathbb {R} .}" loading="lazy"></dd></dl>

Damit kann der Adelering über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> folgendermaßen dargestellt werden:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/647f505973355ae79e9cbf2e442ef54a0c553d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.918ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}" loading="lazy"></dd></dl>

Dies ist ein algebraischer Isomorphismus. Für einen beliebigen <a href="Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper">algebraischen Zahlkörper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> gilt nun:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}\cong \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6fa1c0793006129487d92d1aca07c823fcf4e14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.722ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{K}\cong \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K,}" loading="lazy"></dd></dl>

wobei wir die rechte Seite mit folgender <a href="Topologischer_Raum" title="Topologischer Raum">Topologie</a> versehen. Es gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/badc61e8e6962c3c3ca2c02bb8842b9143665a76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.133ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },}" loading="lazy"> wobei die rechte Seite insgesamt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=[K:\mathbb {Q} ]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n:=[K:\mathbb {Q} ]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/928e1306c4c3c4969eeb7399d4452a0a356297ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.245ex; height:2.843ex;" alt="{\displaystyle n:=[K:\mathbb {Q} ]}" loading="lazy"> Summanden hat. Wir installieren auf der rechten Seite die <a href="Produkttopologie" title="Produkttopologie">Produkttopologie</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {A} _{\mathbb {Q} })^{n}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {A} _{\mathbb {Q} })^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f950b21a0dd57e866d477fc8f5d992d01d33d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.217ex; height:2.843ex;" alt="{\displaystyle (\mathbb {A} _{\mathbb {Q} })^{n}}" loading="lazy"> und transportieren diese mit Hilfe des <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5b838cc1fde05df765d66853421d97c91ece129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.253ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.}" loading="lazy">

Beweis: Wir beweisen zunächst die Gleichung für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }.}">

<msub>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd6fff3543a0e1e541ce3aa9924de11265cfb018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.836ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }.}" loading="lazy"> Es ist also zu zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}">

<msub>

</mrow>

</mrow>

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<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/647f505973355ae79e9cbf2e442ef54a0c553d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.918ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}" loading="lazy"> Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} =({\widehat {\mathbb {Z} }}\times \mathbb {R} )\otimes _{\mathbb {Z} }\mathbb {Q} \cong ({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )\times (\mathbb {R} \otimes _{\mathbb {Z} }\mathbb {Q} )\cong ({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )\times \mathbb {R} ,}">

<msub>

</mrow>

</mrow>

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<msub>

</mrow>

</msub>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} =({\widehat {\mathbb {Z} }}\times \mathbb {R} )\otimes _{\mathbb {Z} }\mathbb {Q} \cong ({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )\times (\mathbb {R} \otimes _{\mathbb {Z} }\mathbb {Q} )\cong ({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )\times \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66f7d215aae366377681f5ffe7d3097f6410afff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.278ex; height:3.343ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} =({\widehat {\mathbb {Z} }}\times \mathbb {R} )\otimes _{\mathbb {Z} }\mathbb {Q} \cong ({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )\times (\mathbb {R} \otimes _{\mathbb {Z} }\mathbb {Q} )\cong ({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )\times \mathbb {R} ,}" loading="lazy"> wobei man das „Ausmultiplizieren“ beim Tensorprodukt durch eine Betrachtung mit Basen einsieht. Die zweite Isomorphie folgt dadurch, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-<a href="Modulhomomorphismus" title="Modulhomomorphismus">lineare Abbildungen</a> bereits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">-linear sind. Offensichtlich reicht es zu zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} }">

<msub>

</mrow>

</mrow>

</mrow>

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<mover>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82a32c6fb0fa55cf596dfa0244ae675bfe405472.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.73ex; height:3.509ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} }" loading="lazy"> ist. Wir rechnen hierzu die universelle Eigenschaft des <a href="Tensorprodukt" title="Tensorprodukt">Tensorproduktes</a> nach. Definiere eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-bilineare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \rightarrow \mathbb {A} _{\mathbb {Q} ,fin}}">

<mover>

</mrow>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \rightarrow \mathbb {A} _{\mathbb {Q} ,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b6c0f9d3894ce32eb54ddaf62c85a2a41fedb80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.662ex; height:3.509ex;" alt="{\displaystyle \Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \rightarrow \mathbb {A} _{\mathbb {Q} ,fin}}" loading="lazy"> via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((a_{p})_{p},q)\mapsto (a_{p}\cdot q)_{p}.}">

<msub>

</mrow>

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<msub>

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<msub>

</mrow>

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((a_{p})_{p},q)\mapsto (a_{p}\cdot q)_{p}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cda7cc8591e7eecd7b5612d109bc28d84593931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.237ex; height:3.009ex;" alt="{\displaystyle ((a_{p})_{p},q)\mapsto (a_{p}\cdot q)_{p}.}" loading="lazy"> Diese Abbildung ist offensichtlich wohldefiniert, da nur endlich viele <a href="Primzahl" title="Primzahl">Primzahlen</a> den <a href="Nenner" class="mw-redirect" title="Nenner">Nenner</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\in \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03aaa9327494402fb81829278f439afcf201313a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.718ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {Q} }" loading="lazy"> teilen. Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-bilinear.

Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> ein weiterer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-Modul mit einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-bilinearen Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \rightarrow Z.}">

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \rightarrow Z.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61d12ceab9e7ace3dbdfa8fa624f3464256e0798.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.463ex; height:3.176ex;" alt="{\displaystyle \phi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \rightarrow Z.}" loading="lazy"> Zu zeigen ist, dass es genau eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-lineare Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta :\mathbb {A} _{\mathbb {Q} ,fin}\rightarrow Z}">

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta :\mathbb {A} _{\mathbb {Q} ,fin}\rightarrow Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87d6baee6b1951645ad05d0040b53af55cd560d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.426ex; height:2.843ex;" alt="{\displaystyle \theta :\mathbb {A} _{\mathbb {Q} ,fin}\rightarrow Z}" loading="lazy"> gibt, mit der Eigenschaft: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \circ \Psi =\phi .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \circ \Psi =\phi .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b86eaf4ef5ddf64c6f1def4a6605f0dfd4615d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.224ex; height:2.509ex;" alt="{\displaystyle \theta \circ \Psi =\phi .}" loading="lazy"> Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> wird wie folgt definiert: Zu gegebenem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{p})_{p}}">

<msub>

</mrow>

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u_{p})_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64c7742b13d0b77afe89a88fb675a990d6b70676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.257ex; height:3.009ex;" alt="{\displaystyle (u_{p})_{p}}" loading="lazy"> existiert ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b27dd9568917d35decc39e59177f1ba9bc0d74a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle u\in \mathbb {N} }" loading="lazy"> und ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{p})_{p}\in {\widehat {\mathbb {Z} }},}">

<msub>

</mrow>

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<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (v_{p})_{p}\in {\widehat {\mathbb {Z} }},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3018aab8c0266607576f0d64d1945a5eceedba0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.093ex; height:3.509ex;" alt="{\displaystyle (v_{p})_{p}\in {\widehat {\mathbb {Z} }},}" loading="lazy"> sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle u_{p}={\frac {1}{u}}\cdot v_{p}}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle u_{p}={\frac {1}{u}}\cdot v_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa69175d3268599e3aef1d7ad1c28748298db0a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.13ex; height:3.343ex;" alt="{\displaystyle \textstyle u_{p}={\frac {1}{u}}\cdot v_{p}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> gilt. Definiere dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \theta ((u_{p})_{p}):=\phi ((v_{p})_{p},{\frac {1}{u}}).}">

<msub>

</mrow>

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<msub>

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<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \theta ((u_{p})_{p}):=\phi ((v_{p})_{p},{\frac {1}{u}}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2916a74c3420e7244e045e8751de0af44f5b6db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.609ex; height:3.343ex;" alt="{\displaystyle \textstyle \theta ((u_{p})_{p}):=\phi ((v_{p})_{p},{\frac {1}{u}}).}" loading="lazy"> Man mache sich klar, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> wohldefiniert ist, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-linear und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \circ \Psi =\phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta \circ \Psi =\phi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4338147e88e265c38ff18902bc8472ee30165743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.577ex; height:2.509ex;" alt="{\displaystyle \theta \circ \Psi =\phi }" loading="lazy"> erfüllt. Weiterhin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> durch diese Eigenschaften bereits eindeutig festgelegt. Der allgemeine Fall kann ähnlich gezeigt werden und wird im folgenden Abschnitt noch allgemeiner bewiesen.

<div class="mw-heading mw-heading2"><h2 id="Der_Adelering_𝔸L_bei_einer_Körpererweiterung_L/K">Der Adelering 𝔸L bei einer Körpererweiterung L/K</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Alternative_Beschreibung_des_Adelerings_𝔸L_im_Fall_L/K">Alternative Beschreibung des Adelerings 𝔸L im Fall L/K</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> eine endliche Körpererweiterung. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein algebraischer Zahlkörper, dann ist die Körpererweiterung <a href="Separable_K%C3%B6rpererweiterung" class="mw-redirect" title="Separable Körpererweiterung">separabel</a>. Im Funktionenkörperfall kann sie ebenfalls als separabel angenommen werden, vgl. <a href="#Literatur">Weil (1967)</a>, S. 48f. Damit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> wieder ein globaler Körper und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b04a9b4c5e625c0a2de5c0699935bb0a09aa79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}}" loading="lazy"> ist definiert. Für eine Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> und eine Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> definiere

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\mid v,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fc67ab52c138707cac00884757d56646ed95270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.376ex; height:2.843ex;" alt="{\displaystyle w\mid v,}" loading="lazy"></dd></dl>

falls der Betrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{w}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{w}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e1e06bad0120d64323d2e8986021edccdd27dde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.382ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{w}}" loading="lazy"> eingeschränkt auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> in der Äquivalenzklasse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> liegt. Man sagt, die Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> liegt über der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb7b87f7afb1452b9d0e287f8ef746a9912c8333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle v.}" loading="lazy"> Definiere nun

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

<msub>

</mrow>

</mrow>

</msub>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L_{v}&:=\prod _{w\mid v}L_{w},\\{\widetilde {{\mathcal {O}}_{v}}}&:=\prod _{w\mid v}{\mathcal {O}}_{w}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25e863403af63b3bc3ec45b1d0ccb548621579f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:14.639ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}L_{v}&:=\prod _{w\mid v}L_{w},\\{\widetilde {{\mathcal {O}}_{v}}}&:=\prod _{w\mid v}{\mathcal {O}}_{w}.\end{aligned}}}" loading="lazy"></dd></dl>

Beachte, dass mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> die Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> die Stellen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> bezeichnet werden. Beachte weiterhin, dass beide Produkte endlich sind.

Bemerkung: Man kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{w}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{w}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f1f877323bcb0fab941ee2d2c30ddaa045ab844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.992ex; height:2.509ex;" alt="{\displaystyle L_{w}}" loading="lazy"> einbetten, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> liegt. Dadurch kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> diagonal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d3252ec5dbd18467fc109498966e5abb43123b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle L_{v}}" loading="lazy"> einbetten und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d3252ec5dbd18467fc109498966e5abb43123b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle L_{v}}" loading="lazy"> wird dadurch eine kommutative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy">-Algebra vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{w\mid v}[L_{w}:K_{v}]=[L:K].}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{w\mid v}[L_{w}:K_{v}]=[L:K].}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec65d76757e32639b9425f8a55597b6542217ff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.968ex; height:3.343ex;" alt="{\displaystyle \textstyle \sum _{w\mid v}[L_{w}:K_{v}]=[L:K].}" loading="lazy">

Es gilt nun

<msub>

</mrow>

</mrow>

</msub>

<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<mover>

<msub>

</mrow>

</mrow>

</msub>

</mover>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}={\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}L_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e832624cf2f4e87e4ad69a1670af1abd72b41acd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.709ex; height:7.176ex;" alt="{\displaystyle \mathbb {A} _{L}={\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}L_{v}.}" loading="lazy"></dd></dl>

Der Beweis beruht auf elementaren Eigenschaften restringierter Produkte.

Der Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> kann dabei wie folgt kanonisch in den Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> eingebettet werden: Dem Adel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{v})_{v}\in \mathbb {A} _{K}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=(a_{v})_{v}\in \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba5cb747859cb9b26ae439c69253f3480fee800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:2.843ex;" alt="{\displaystyle a=(a_{v})_{v}\in \mathbb {A} _{K}}" loading="lazy"> wird das Adel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'=(a'_{w})_{w}\in \mathbb {A} _{L}}">

<msup>

</msup>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a'=(a'_{w})_{w}\in \mathbb {A} _{L}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b5c505612c040afe970c6c0a1b05e6b94c1ec84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.74ex; height:3.009ex;" alt="{\displaystyle a'=(a'_{w})_{w}\in \mathbb {A} _{L}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a'_{w}=a_{v}\in K_{v}\subset L_{w}}">

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle a'_{w}=a_{v}\in K_{v}\subset L_{w}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1976de0c2ed19d2db78cdb6bee39ecdfb6a1e607.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.93ex; height:2.509ex;" alt="{\displaystyle a'_{w}=a_{v}\in K_{v}\subset L_{w}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\mid v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\mid v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a4afc1adbbeb4a3da5c75229ecf1f45b0db567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.729ex; height:2.843ex;" alt="{\displaystyle w\mid v}" loading="lazy"> zugeordnet. Deshalb kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> als Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b04a9b4c5e625c0a2de5c0699935bb0a09aa79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}}" loading="lazy"> aufgefasst werden. Ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(a_{w})_{w}\in \mathbb {A} _{L}}">

<msub>

</mrow>

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<msub>

</mrow>

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<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=(a_{w})_{w}\in \mathbb {A} _{L}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ab5cb6c30efb8f5181f5fe7dac5830d6d26f7ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.056ex; height:2.843ex;" alt="{\displaystyle a=(a_{w})_{w}\in \mathbb {A} _{L}}" loading="lazy"> liegt also genau dann in der Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K},}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bef6ca8bc0051c81dee17fa2d17e74c6029d22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K},}" loading="lazy"> wenn seine Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{w}\in K_{v}}">

<msub>

</mrow>

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<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{w}\in K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13373221f3793ac74e082990c28844de5d645bf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.482ex; height:2.509ex;" alt="{\displaystyle a_{w}\in K_{v}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\mid v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\mid v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a4afc1adbbeb4a3da5c75229ecf1f45b0db567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.729ex; height:2.843ex;" alt="{\displaystyle w\mid v}" loading="lazy"> erfüllen und weiterhin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{w}=a_{w'}}">

<msub>

</mrow>

</msub>

<msub>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{w}=a_{w'}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5dcd528f67d25ec92be4d5e2c970b732fa4e3ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.908ex; height:2.009ex;" alt="{\displaystyle a_{w}=a_{w'}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\mid v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\mid v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a4afc1adbbeb4a3da5c75229ecf1f45b0db567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.729ex; height:2.843ex;" alt="{\displaystyle w\mid v}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w'\mid v}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w'\mid v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cce58979bf4e4247733005ccf71cbd0f3738d0b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.414ex; height:3.009ex;" alt="{\displaystyle w'\mid v}" loading="lazy"> für die gleiche Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> gilt.

<div class="mw-heading mw-heading3"><h3 id="Der_Adelering_𝔸L_als_Tensorprodukt">Der Adelering 𝔸L als Tensorprodukt</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> eine endliche Körpererweiterung. Dann gilt:

<msub>

</mrow>

</mrow>

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<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\otimes _{K}L.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/960e517f3ccd1cf0c28ceed9aa54766b2f30937e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.262ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\otimes _{K}L.}" loading="lazy"></dd></dl>

Dies ist ein algebraischer und topologischer Isomorphismus, wobei wir die Topologie des Tensorproduktes analog wie in dem <a href="#Alternative_Definition_des_Adelerings_eines_Zahlkörpers">Lemma über die alternative Definition des Adelerings eines Zahlkörpers</a> konstruieren. Um dies zu tun, ist es wichtig, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [L:K]<\infty .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [L:K]<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7d4633ab1548c3a426f2377c28ce88dc47e9ba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.949ex; height:2.843ex;" alt="{\displaystyle [L:K]<\infty .}" loading="lazy"> Mit der Hilfe dieses Isomorphismus, ist die Inklusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}\subset \mathbb {A} _{L}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}\subset \mathbb {A} _{L}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1d33d9f0a4b6ad70a38184d8e5dfd822da1d301.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.499ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}\subset \mathbb {A} _{L}}" loading="lazy"> durch die Funktion

<mtr>

<mtd>

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</mrow>

</mrow>

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</mtd>

<mtd>

<msub>

</mrow>

</mrow>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {A} _{K}&\hookrightarrow \mathbb {A} _{L},\\\alpha &\mapsto \alpha \otimes _{K}1.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/543f31dcc0ec497072ac12cbefa5a639fd1971c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.567ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathbb {A} _{K}&\hookrightarrow \mathbb {A} _{L},\\\alpha &\mapsto \alpha \otimes _{K}1.\end{aligned}}}" loading="lazy"></dd></dl>

Darüber hinaus können die Hauptadele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> mit einer Untergruppe der Hauptadele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> identifiziert werden via der Abbildung

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

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<mtd>

<msub>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K&\hookrightarrow (K\otimes _{K}L)\cong L,\\\alpha &\mapsto 1\otimes _{K}\alpha .\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e316a0b1c4f6d23d6d647216ebc74b64cb031384.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.044ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}K&\hookrightarrow (K\otimes _{K}L)\cong L,\\\alpha &\mapsto 1\otimes _{K}\alpha .\end{aligned}}}" loading="lazy"></dd></dl>

Beweis: Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1},\dotsc ,\omega _{n}}">

<msub>

</mrow>

</msub>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1},\dotsc ,\omega _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c6ad6c0cb4a686eaf0c995f611d4acf59a677d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.343ex; height:2.009ex;" alt="{\displaystyle \omega _{1},\dotsc ,\omega _{n}}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d6358b7808f1224473a219f51e5eede2495fa88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle L.}" loading="lazy"> Es gilt nun, dass

<mover>

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</mrow>

</mrow>

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</mover>

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<msub>

</mrow>

</mrow>

</msub>

<msub>

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<msub>

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<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}\cong {\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0983c00cd822fee7db2033460f88fcdf0703ca75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.305ex; height:3.176ex;" alt="{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}\cong {\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n}}" loading="lazy"></dd></dl>

für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7ba401aee0189f8031d21020a0c640a03339c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.774ex; height:2.009ex;" alt="{\displaystyle v,}" loading="lazy"> vgl. <a href="#Literatur">Cassels (1967)</a>, S. 61.

Wir haben einen kanonischen Isomorphismus:

<mtr>

<mtd>

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</mrow>

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</msub>

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

</mpadded>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

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</msub>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n}\cong K_{v}\otimes _{K}L&{\xrightarrow {\cong }}L_{v}=\prod _{w\mid v}L_{w},\\\alpha _{v}\otimes a&\mapsto (\alpha _{v}\cdot (\tau _{w}(a)))_{w}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5facaee7f4ab83c81aad3295628ad448eb00d85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; margin-top: -0.289ex; width:50.845ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n}\cong K_{v}\otimes _{K}L&\xrightarrow {\cong } L_{v}=\prod _{w\mid v}L_{w},\\\alpha _{v}\otimes a&\mapsto (\alpha _{v}\cdot (\tau _{w}(a)))_{w}\end{aligned}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{w}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{w}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e28eafeae6c69bc96fe93b3d773cee7a7727f537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.425ex; height:2.009ex;" alt="{\displaystyle \tau _{w}}" loading="lazy"> die kanonische Einbettung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{w}:L\rightarrow L_{w}}">

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau _{w}:L\rightarrow L_{w}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d815fcd06ca0cb0d84744372c51b51b41ab10820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.551ex; height:2.509ex;" alt="{\displaystyle \tau _{w}:L\rightarrow L_{w}}" loading="lazy"> ist und wie üblich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\mid v}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a4afc1adbbeb4a3da5c75229ecf1f45b0db567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.729ex; height:2.843ex;" alt="{\displaystyle w\mid v}" loading="lazy"> gilt. Indem wir auf beiden Seiten das restringierte Produkt mit Restriktionsbedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/478979d80dd03bef58158626abeb1c9a4c3a8bcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:3.176ex;" alt="{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}}" loading="lazy"> bilden, folgt die Behauptung:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {A} _{K}\otimes _{K}L&=({\widehat {\prod \limits _{v}}}^{{\mathcal {O}}_{v}}K_{v})\otimes _{K}L\\&\cong {\widehat {\prod \limits _{v}}}^{({\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n})}(K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n})\\&\cong {\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}(K_{v}\otimes _{K}L)\\&\cong {\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}L_{v}\\&=\mathbb {A} _{L}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3bee43e9d279c1a12f7c18c2ba8990004461f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.005ex; width:51.765ex; height:31.176ex;" alt="{\displaystyle {\begin{aligned}\mathbb {A} _{K}\otimes _{K}L&=({\widehat {\prod \limits _{v}}}^{{\mathcal {O}}_{v}}K_{v})\otimes _{K}L\\&\cong {\widehat {\prod \limits _{v}}}^{({\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n})}(K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n})\\&\cong {\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}(K_{v}\otimes _{K}L)\\&\cong {\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}L_{v}\\&=\mathbb {A} _{L}.\end{aligned}}}" loading="lazy"></dd></dl>

Dieser Beweis findet sich in <a href="#Literatur">Cassels (1967)</a>, S. 65.

Korollar: Der Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> als additive Gruppe

Als additive Gruppe betrachtet gilt:

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\oplus \dotsc \oplus \mathbb {A} _{K},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2194b01c06fd61f9351605b5ecc21c301295a747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.921ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\oplus \dotsc \oplus \mathbb {A} _{K},}" loading="lazy"></dd></dl>

wobei die linke Seite insgesamt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:=[L:K]}">

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<annotation encoding="application/x-tex">{\displaystyle n:=[L:K]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0125a16d06197f8f42d1c6d02b17e6a6c3fdf8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.02ex; height:2.843ex;" alt="{\displaystyle n:=[L:K]}" loading="lazy"> Summanden hat.

Die Hauptadele von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b04a9b4c5e625c0a2de5c0699935bb0a09aa79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}}" loading="lazy"> gehen dabei auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\oplus \dotsc \oplus K,}">

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<annotation encoding="application/x-tex">{\displaystyle K\oplus \dotsc \oplus K,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9da91033144b91fe3ebb26d574f6298ff2959f20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.183ex; height:2.509ex;" alt="{\displaystyle K\oplus \dotsc \oplus K,}" loading="lazy"> wobei hier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> als Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> aufgefasst wird. Die Summe hat dabei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Summanden.

<div class="mw-heading mw-heading2"><h2 id="Definition_des_Adelerings_eines_K-Vektorraums_E_und_einer_K-Algebra_A">Definition des Adelerings eines K-Vektorraums E und einer K-Algebra A</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Alternative_Beschreibung_des_Adelerings_eines_globalen_Körpers">Alternative Beschreibung des Adelerings eines globalen Körpers</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> eine endliche Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

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<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> umfasst. Hierbei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> die unendlichen Stellen des globalen Körpers. Definiere

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P):=\prod _{v\in P}K_{v}\times \prod _{v\notin P}{\mathcal {O}}_{v}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77d33237d20ee742cf24e0039828aeecab9a9823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.07ex; height:6.009ex;" alt="{\displaystyle \mathbb {A} _{K}(P):=\prod _{v\in P}K_{v}\times \prod _{v\notin P}{\mathcal {O}}_{v}.}" loading="lazy"></dd></dl>

Man definiert die Addition und Multiplikation komponentenweise und versieht den entstandenen Ring mit der Produkttopologie. Es entsteht ein lokalkompakter, topologischer Ring. Anders formuliert: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}(P)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/808d2fc4d29adfac554e6501d30479c4511b6553.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.926ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}(P)}" loading="lazy"> ist die Menge aller <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x=(x_{v})_{v}\in \prod _{v}K_{v},}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle x=(x_{v})_{v}\in \prod _{v}K_{v},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97abff8493a785bf085d6f7b02b137277cc8ec90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.727ex; height:3.009ex;" alt="{\displaystyle \textstyle x=(x_{v})_{v}\in \prod _{v}K_{v},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{v}\in {\mathcal {O}}_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{v}\in {\mathcal {O}}_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5d72637f8f96a46865977c20ad0b241c46bb91e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.079ex; height:2.509ex;" alt="{\displaystyle x_{v}\in {\mathcal {O}}_{v}}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\notin P,}">

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<annotation encoding="application/x-tex">{\displaystyle v\notin P,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/239dc3fefb584ceb89dc7b88573d2489c71251d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.676ex;" alt="{\displaystyle v\notin P,}" loading="lazy"> also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{v}|_{v}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle |x_{v}|_{v}\leq 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7d23ff8f5f90447f074e0c73e472d313aa6b918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.943ex; height:3.009ex;" alt="{\displaystyle |x_{v}|_{v}\leq 1}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\notin P,}">

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<annotation encoding="application/x-tex">{\displaystyle v\notin P,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/239dc3fefb584ceb89dc7b88573d2489c71251d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.676ex;" alt="{\displaystyle v\notin P,}" loading="lazy"> gelten soll.

Bemerkung: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

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<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy"> eine weitere endliche Teilmenge der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> mit der Eigenschaft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\subset P',}">

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<annotation encoding="application/x-tex">{\displaystyle P\subset P',}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89370fb5181d716b0ba46d5c37753675646d8a02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.997ex; height:2.843ex;" alt="{\displaystyle P\subset P',}" loading="lazy"> dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}(P)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/808d2fc4d29adfac554e6501d30479c4511b6553.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.926ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}(P)}" loading="lazy"> ein offener Unterring von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}(P').}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P').}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a6654169cb036ef60a3a0119bdf401059f420.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.334ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{K}(P').}" loading="lazy">

Wir geben nun eine alternative Definition des Adelerings. Mengentheoretisch ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> die Vereinigung über alle Mengen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}(P),}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4db6b1b410fe7ace67e9a306aeb3655c5ac255b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.573ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}(P),}" loading="lazy"> wobei die Vereinigung alle endlichen Teilmengen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\supset P_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\supset P_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20bea132ae42393e3ec7b821a6015e3c3cbc1b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.212ex; height:2.509ex;" alt="{\displaystyle P\supset P_{\infty }}" loading="lazy"> von der gesamten Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> durchläuft. Es gilt also

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mtext> endlich </mtext>

</mrow>

</mfrac>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}=\bigcup _{P\supset P_{\infty }, \atop P{\text{ endlich }}}\mathbb {A} _{K}(P).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03be0c57a76baa07fb0d41d5efac88397adecea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:21.304ex; height:7.009ex;" alt="{\displaystyle \mathbb {A} _{K}=\bigcup _{P\supset P_{\infty }, \atop P{\text{ endlich }}}\mathbb {A} _{K}(P).}" loading="lazy"></dd></dl>

In anderen Worten ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> nichts anderes als die Menge aller <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{v})_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=(x_{v})_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ebc5ff290b926ced6ae58e2a55f04229c99727a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.626ex; height:2.843ex;" alt="{\displaystyle x=(x_{v})_{v}}" loading="lazy"> für die gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{v}|_{v}\leq 1}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x_{v}|_{v}\leq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7d23ff8f5f90447f074e0c73e472d313aa6b918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.943ex; height:3.009ex;" alt="{\displaystyle |x_{v}|_{v}\leq 1}" loading="lazy"> für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v<\infty .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ddaafee28d99862569bfea91d3f2701d31c79b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.197ex; height:1.843ex;" alt="{\displaystyle v<\infty .}" loading="lazy"> Die Topologie auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> wird so definiert, dass alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}(P)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/808d2fc4d29adfac554e6501d30479c4511b6553.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.926ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}(P)}" loading="lazy"> offene Unterringe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> werden sollen. Dadurch wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> ein lokalkompakter, topologischer Ring.

Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> eine Stelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> eine endliche Teilmenge der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> welche die unendlichen Stellen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> enthält. Es gilt:

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P)=\prod _{w\in P}K_{w}\times \prod _{w\notin P}{\mathcal {O}}_{w}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fcce35fef9c4804ce3894acdd6e07c4f5ba90dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.941ex; height:6.009ex;" alt="{\displaystyle \mathbb {A} _{K}(P)=\prod _{w\in P}K_{w}\times \prod _{w\notin P}{\mathcal {O}}_{w}.}" loading="lazy"></dd></dl>

Definiere nun

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}'(P,v):=\prod _{w\in P\setminus \{v\}}K_{w}\times \prod _{w\notin P}{\mathcal {O}}_{w}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb367e86a47bf7f95efed324206cf2a5e01b1aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.013ex; height:6.009ex;" alt="{\displaystyle \mathbb {A} _{K}'(P,v):=\prod _{w\in P\setminus \{v\}}K_{w}\times \prod _{w\notin P}{\mathcal {O}}_{w}.}" loading="lazy"></dd></dl>

Dann gilt:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P)\cong K_{v}\times \mathbb {A} _{K}'(P,v).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/357dabd62bcd65385403aeb1b75f06a375851164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.602ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{K}(P)\cong K_{v}\times \mathbb {A} _{K}'(P,v).}" loading="lazy"></dd></dl>

Definiere weiterhin:

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}'(v):=\bigcup _{P\supset (P_{\infty }\cup \{v\})}\mathbb {A} _{K}'(P,v),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/befedcf3594000d2413f0f1e290e53cfcf0b1104.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.058ex; height:6.009ex;" alt="{\displaystyle \mathbb {A} _{K}'(v):=\bigcup _{P\supset (P_{\infty }\cup \{v\})}\mathbb {A} _{K}'(P,v),}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> alle endlichen Teilmengen der Stellenmenge durchläuft, welche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }\cup \{v\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\infty }\cup \{v\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69da05530bc999c12546c580870d8aa3d966b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.403ex; height:2.843ex;" alt="{\displaystyle P_{\infty }\cup \{v\}}" loading="lazy"> enthält. Dann gilt offensichtlich:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}\cong K_{v}\times \mathbb {A} _{K}'(v),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57205af77429c05de343ee7facdcaae75341488e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.268ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{K}\cong K_{v}\times \mathbb {A} _{K}'(v),}" loading="lazy"></dd></dl>

via der Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{w})_{w}\mapsto (a_{v},(a_{w})_{w\neq v}).}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{w})_{w}\mapsto (a_{v},(a_{w})_{w\neq v}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31a21ddabf75b4837440c5fc595642dbb39dc911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.154ex; height:3.009ex;" alt="{\displaystyle (a_{w})_{w}\mapsto (a_{v},(a_{w})_{w\neq v}).}" loading="lazy"> Dies kann mit jeder endlichen Stellenmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {P}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {P}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00ff2ce2da00b0161cf17cf5565cf1c886197a82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:2.676ex;" alt="{\displaystyle {\widetilde {P}}}" loading="lazy"> anstelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{v\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bd92cd3b725703bb9ac92a362253b9732ed7b30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.453ex; height:2.843ex;" alt="{\displaystyle \{v\}}" loading="lazy"> ebenso gemacht werden.

Mit Hilfe der obigen Definition von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}'(v)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}'(v)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/690ac2c3d364c5fc9ee51b588ffbffba8631f4a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.308ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{K}'(v)}" loading="lazy"> gibt es eine natürliche Einbettung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}\hookrightarrow \mathbb {A} _{K}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}\hookrightarrow \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a266321c1fed6915ea46a175e693f7b89f40b956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.281ex; height:2.509ex;" alt="{\displaystyle K_{v}\hookrightarrow \mathbb {A} _{K}}" loading="lazy"> und eine natürliche Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}\twoheadrightarrow K_{v}.}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}\twoheadrightarrow K_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4043e015fde33c116b718af68c4fedaea567124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.635ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}\twoheadrightarrow K_{v}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Der_Adelering_eines_K-Vektorraums_E">Der Adelering eines K-Vektorraums E</h3></div>

Die folgenden beiden Definitionen orientieren sich an <a href="#Literatur">Weil (1967)</a>, S. 60ff. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> wie bisher ein globaler Körper und sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensionaler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Vektorraum, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<\infty .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5067ba3a25c287a7791babec602283f3668a356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.464ex; height:1.843ex;" alt="{\displaystyle n<\infty .}" loading="lazy"> Wir fixieren eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1},\dotsc ,\omega _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1},\dotsc ,\omega _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c6ad6c0cb4a686eaf0c995f611d4acf59a677d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.343ex; height:2.009ex;" alt="{\displaystyle \omega _{1},\dotsc ,\omega _{n}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a2566d01f104ef084ea424b8b35c2534f7f902b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}" loading="lazy"> Für jede Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> schreiben wir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{v}:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{v}:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d46435fa2307be2557ddaf89eb1f8953f4b01997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.474ex; height:2.509ex;" alt="{\displaystyle E_{v}:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \dotsc \oplus K_{v}\omega _{n}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}:={\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n}.}">

<mover>

<msub>

</mrow>

</mrow>

</msub>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}:={\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f168f302dc5e76597d2481e74703323257adffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.599ex; height:3.176ex;" alt="{\displaystyle {\widetilde {{\mathcal {O}}_{v}}}:={\mathcal {O}}_{v}\omega _{1}\oplus \dotsc \oplus {\mathcal {O}}_{v}\omega _{n}.}" loading="lazy"> Definiere dann den Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> als

<msub>

</mrow>

</mrow>

</msub>

<msup>

<mover>

</mrow>

</munder>

</mover>

</mrow>

<mover>

<msub>

</mrow>

</mrow>

</msub>

</mover>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}:={\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}E_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9f01ad9086513745e2098f8033d79c10f74705c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.624ex; height:7.176ex;" alt="{\displaystyle \mathbb {A} _{E}:={\widehat {\prod \limits _{v}}}^{\widetilde {{\mathcal {O}}_{v}}}E_{v}.}" loading="lazy"></dd></dl>

Diese Definition ist angelehnt an die alternative Beschreibung des Adelerings als Tensorprodukt. Wir konstruieren wieder die Topologie auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449107507b1283251850665759926442312e9722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.745ex; height:2.509ex;" alt="{\displaystyle E_{v}}" loading="lazy"> analog wie in dem <a href="#Alternative_Definition_des_Adelerings_eines_Zahlkörpers">Lemma über die alternative Definition des Adelerings eines Zahlkörpers</a>. Um dies zu tun, ist es wichtig, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} _{K}(E)=n<\infty .}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} _{K}(E)=n<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cf8206074acfcb372c44ba3f55d0d6c15c042f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.715ex; height:2.843ex;" alt="{\displaystyle \operatorname {dim} _{K}(E)=n<\infty .}" loading="lazy"> Wir versehen dann den Adelering von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> mit der restringierten Produkttopologie.

Analog wie in dem Abschnitt über den Adelering bei einer Körpererweiterung erhalten wir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}=E\otimes _{K}\mathbb {A} _{K}.}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}=E\otimes _{K}\mathbb {A} _{K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3451640d1fb248852c347b21b9c24669a14c442e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.591ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}=E\otimes _{K}\mathbb {A} _{K}.}" loading="lazy"> Dann kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\mapsto e\otimes 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\mapsto e\otimes 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57e58bc539a3d6cba844457b7fe5794760101667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.784ex; height:2.343ex;" alt="{\displaystyle e\mapsto e\otimes 1}" loading="lazy"> natürlich in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> eingebettet werden.

Im Folgenden wird eine alternative Definition der Topologie auf dem Adelering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> gegeben. Die Topologie auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> ist gegeben als die gröbste Topologie, für welche die Linearformen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89862747e88ca143e979241a9a243b5ef66ddc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.509ex;" alt="{\displaystyle E,}" loading="lazy"> das sind lineare Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda :E\rightarrow K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda :E\rightarrow K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/163277cc52d24227f4e4e35ea715b011e80f76a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.395ex; height:2.509ex;" alt="{\displaystyle \lambda :E\rightarrow K,}" loading="lazy"> die ausgedehnt werden zu linearen Abbildungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K},}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bef6ca8bc0051c81dee17fa2d17e74c6029d22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K},}" loading="lazy"> stetig sind. Man benutzt jeweils, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> auf natürliche Art und Weise in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> eingebettet werden können. Mit anderen Worten: Die Wahl einer Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> liefert einen Isomorphismus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d63366b3d00300e06eee81786182062b98775c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle K^{n}}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89862747e88ca143e979241a9a243b5ef66ddc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.509ex;" alt="{\displaystyle E,}" loading="lazy"> also einen Isomorphismus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {A} _{K})^{n}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {A} _{K})^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abc112d850a0ed332f959609c52845b2ff10c8e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.399ex; height:2.843ex;" alt="{\displaystyle (\mathbb {A} _{K})^{n}}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ccdb1ba27215e0ae8d8fc61df305b7d4b90073f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.813ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}.}" loading="lazy"> Man kann nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {A} _{K})^{n}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {A} _{K})^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abc112d850a0ed332f959609c52845b2ff10c8e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.399ex; height:2.843ex;" alt="{\displaystyle (\mathbb {A} _{K})^{n}}" loading="lazy"> mit der Produkttopologie versehen und diese mit Hilfe des Isomorphismus nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> transportieren. Die Wahl der Topologie hängt nicht von der Wahl der Basis ab, denn eine weitere Basiswahl definiert einen zweiten Isomorphismus. Die Komposition der Isomorphismen ergibt einen linearen Homöomorphismus, der die eine Topologie in die andere überführt. Man kann dies wie folgt darstellen:

<mtr>

<mtd>

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</mrow>

</mrow>

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</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \dotsc \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \dotsc \oplus \mathbb {A} _{K}\\&=(\mathbb {A} _{K})^{n},\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/446b7b4700a0b1eb56561e4fef326aba177acd5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:38.98ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \dotsc \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \dotsc \oplus \mathbb {A} _{K}\\&=(\mathbb {A} _{K})^{n},\end{aligned}}}" loading="lazy"></dd></dl>

wobei die auftretenden Summen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Summanden haben. Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=L,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=L,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b496246a197204476973338e8b5c5f530dd48685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.104ex; height:2.509ex;" alt="{\displaystyle E=L,}" loading="lazy"> so liefert obige Definition den bereits definierten Adelering.

<div class="mw-heading mw-heading3"><h3 id="Der_Adelering_einer_K-Algebra_A">Der Adelering einer K-Algebra A</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> eine endlichdimensionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Algebra. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> insbesondere ein endlichdimensionaler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Vektorraum. Folglich ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db6915d4a6dd197505d2b64b9cc85d1e6d260d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}}" loading="lazy"> definiert, vgl. dazu den letzten Abschnitt. Wir dehnen die Multiplikation von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db6915d4a6dd197505d2b64b9cc85d1e6d260d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}}" loading="lazy"> aus. Dies geht wie folgt:

Es gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}\cong \mathbb {A} _{K}\otimes _{K}A.}">

<msub>

</mrow>

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<msub>

</mrow>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}\cong \mathbb {A} _{K}\otimes _{K}A.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8b81c7013a8e9372d637c010c7fd68dc3c7260d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.536ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}\cong \mathbb {A} _{K}\otimes _{K}A.}" loading="lazy"> Da wir eine Multiplikation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> und auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> haben, können wir eine Multiplikation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db6915d4a6dd197505d2b64b9cc85d1e6d260d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}}" loading="lazy"> definieren, via

<msub>

</mrow>

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<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\otimes _{K}b)\cdot (c\otimes _{K}d):=(ac)\otimes _{K}(bd)\quad \forall a,c\in \mathbb {A} _{K}{\text{ und }}\forall b,d\in A.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3ffb2c31f5b4d49d2d857a626057b0e416089c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.069ex; height:2.843ex;" alt="{\displaystyle (a\otimes _{K}b)\cdot (c\otimes _{K}d):=(ac)\otimes _{K}(bd)\quad \forall a,c\in \mathbb {A} _{K}{\text{ und }}\forall b,d\in A.}" loading="lazy"></dd></dl>

Alternativ, kann man eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\dotsc ,\alpha _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\dotsc ,\alpha _{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef2882d5782b040c95430bf8dce8ddef16f22a8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.426ex; height:2.009ex;" alt="{\displaystyle \alpha _{1},\dotsc ,\alpha _{n}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> fixieren. Um die Multiplikation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> vollständig zu beschreiben, genügt es zu wissen, wie die Basiselemente miteinander multipliziert werden. Es existieren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i,j,k}\in K,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{i,j,k}\in K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b69c2a685f0a1edd7f77909f11887836f12756dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle \beta _{i,j,k}\in K,}" loading="lazy"> so dass

<msub>

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<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{i}\alpha _{j}=\sum _{k=1}^{n}\beta _{i,j,k}\alpha _{k}\quad \forall i,j.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b19a345864ecd0b3dd656e33ce9c9ccc0b23216f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.722ex; height:6.843ex;" alt="{\displaystyle \alpha _{i}\alpha _{j}=\sum _{k=1}^{n}\beta _{i,j,k}\alpha _{k}\quad \forall i,j.}" loading="lazy"></dd></dl>

Mit Hilfe dieser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i,j,k},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{i,j,k},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d265f561c34c94b224ef6195d21201b75fbc700a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.211ex; height:2.843ex;" alt="{\displaystyle \beta _{i,j,k},}" loading="lazy"> können wir eine Multiplikation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{n}\cong A}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{n}\cong A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ca320d57995f35df8953ab3f3c770261706fe36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.154ex; height:2.343ex;" alt="{\displaystyle K^{n}\cong A}" loading="lazy"> definieren:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{i}e_{j}:=\sum _{k=1}^{n}\beta _{i,j,k}e_{k}\quad \forall i,j.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029df993ca005c769c37d5bfc657b49bd719c0a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.157ex; height:6.843ex;" alt="{\displaystyle e_{i}e_{j}:=\sum _{k=1}^{n}\beta _{i,j,k}e_{k}\quad \forall i,j.}" loading="lazy"></dd></dl>

Und ebenso eine Multiplikation auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K_{v})^{n}\cong A_{v}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (K_{v})^{n}\cong A_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c15957f60a0e9ec9c000dbacaf4e10edcaf60e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.902ex; height:2.843ex;" alt="{\displaystyle (K_{v})^{n}\cong A_{v}}" loading="lazy"> und damit auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {A} _{K})^{n}\cong \mathbb {A} _{A}.}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {A} _{K})^{n}\cong \mathbb {A} _{A}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b4a7955c01a62f6aa61cc425d711b5857bd84cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.287ex; height:2.843ex;" alt="{\displaystyle (\mathbb {A} _{K})^{n}\cong \mathbb {A} _{A}.}" loading="lazy">

Es folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db6915d4a6dd197505d2b64b9cc85d1e6d260d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}}" loading="lazy"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy">-Algebra mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> ist. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> eine endliche Teilmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2746026864cc5896e3e52443a1c917be2df9d8ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}" loading="lazy"> welche eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> enthält. Für jede endliche Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> nenne <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/428b8c2b694e64a025c405eabf5205d6552acd1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.517ex; height:2.009ex;" alt="{\displaystyle \alpha _{v}}" loading="lazy"> das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4714e4b7795af9e2c3d727be09bd091a4d1076.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{v}}" loading="lazy">-Modul erzeugt von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{v}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa03a623215a433047b92c795c1ecbdefd55dc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.42ex; height:2.509ex;" alt="{\displaystyle A_{v}.}" loading="lazy"> Für jede endliche Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> welche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> enthält, definiere

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}(P,\alpha )=\prod \limits _{v\in P}A_{v}\times \prod \limits _{v\notin P}\alpha _{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb47623902c9f1d0349a3bf6656bfce623222d7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:28.125ex; height:6.009ex;" alt="{\displaystyle \mathbb {A} _{A}(P,\alpha )=\prod \limits _{v\in P}A_{v}\times \prod \limits _{v\notin P}\alpha _{v}.}" loading="lazy"></dd></dl>

Man kann zeigen, dass es dann eine endliche Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/671bd891701e0d6cfa6da0114a5dd64233b58709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}" loading="lazy"> gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}(P,\alpha )}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}(P,\alpha )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/423ac83f3ade9c76587929432e2142195bf92a0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.219ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{A}(P,\alpha )}" loading="lazy"> ein offener Unterring von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db6915d4a6dd197505d2b64b9cc85d1e6d260d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}}" loading="lazy"> ist, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\supset P_{0}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\supset P_{0}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29e02ecee0ff3d7e19530900dc4cd750501814eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.037ex; height:2.509ex;" alt="{\displaystyle P\supset P_{0}.}" loading="lazy"> Es gilt dann weiterhin, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db6915d4a6dd197505d2b64b9cc85d1e6d260d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{A}}" loading="lazy"> die Vereinigung aller dieser Unterringe ist. Man kann zeigen, dass im Falle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb6144a97dee4058affe2ed89ee536c3893fe65a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.908ex; height:2.176ex;" alt="{\displaystyle A=K}" loading="lazy"> der oben definierte Adelering kanonisch isomorph zur „ersten“ Definition des Adelerings ist.

<div class="mw-heading mw-heading2"><h2 id="Spur_und_Norm_auf_dem_Adelering">Spur und Norm auf dem Adelering</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Spur_und_Norm_auf_dem_Adelering_2">Spur und Norm auf dem Adelering</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> eine endliche Körpererweiterung des globalen Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Dann gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}=\mathbb {A} _{K}\otimes _{K}L.}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}=\mathbb {A} _{K}\otimes _{K}L.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/827208ea71c558d3d885c7699625f3ef96357dd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.262ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}=\mathbb {A} _{K}\otimes _{K}L.}" loading="lazy"> Mit der Identifikation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K}\otimes _{K}K}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K}\otimes _{K}K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5084b74dee617b0a98eeb8e5ce8a852921f409d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.44ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K}\otimes _{K}K}" loading="lazy"> folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> als abgeschlossener Unterring von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b04a9b4c5e625c0a2de5c0699935bb0a09aa79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}}" loading="lazy"> aufgefasst werden kann. Schreibe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {con} _{L/K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {con} _{L/K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17635bdfc9ba679a32e30b7505d5b5ac2fa5ec19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.122ex; height:2.509ex;" alt="{\displaystyle \operatorname {con} _{L/K}}" loading="lazy"> für diese Einbettung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a6fac61cb834b8ae2686704ba55921b8230f557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.676ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}.}" loading="lazy"> Explizit gilt: Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {A} _{K}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {A} _{K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8ba5dd1cbc34f7b8a23fe53b0d9a100ac1532f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.346ex; height:2.509ex;" alt="{\displaystyle \alpha \in \mathbb {A} _{K}.}" loading="lazy"> Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {con} _{L/K}(\alpha ))_{w}=\alpha _{v}\in K_{v},}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\operatorname {con} _{L/K}(\alpha ))_{w}=\alpha _{v}\in K_{v},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf221721abb62f519e8d1b9d554859505347a45d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.743ex; height:3.176ex;" alt="{\displaystyle (\operatorname {con} _{L/K}(\alpha ))_{w}=\alpha _{v}\in K_{v},}" loading="lazy"> wobei dies für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> gilt.

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M/L/K}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f471b9ce64aa342188c5748f9ae970527384c3b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.416ex; height:2.843ex;" alt="{\displaystyle M/L/K}" loading="lazy"> ein <a href="K%C3%B6rperturm" title="Körperturm">Körperturm</a> globaler Körper. Dann gilt

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {con} _{M/K}(\alpha )=\operatorname {con} _{M/L}(\operatorname {con} _{L/K}(\alpha ))\qquad \forall \alpha \in \mathbb {A} _{K}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8ea0ebf87767e88762c288897a5bc12511ff194.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:48.024ex; height:3.176ex;" alt="{\displaystyle \operatorname {con} _{M/K}(\alpha )=\operatorname {con} _{M/L}(\operatorname {con} _{L/K}(\alpha ))\qquad \forall \alpha \in \mathbb {A} _{K}.}" loading="lazy"></dd></dl>

Schränken wir die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {con} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f3f6e3c6d6aafcf4400fcef54a06beef3a7d3e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:1.676ex;" alt="{\displaystyle \operatorname {con} }" loading="lazy"> auf die Menge der Hauptadele ein, so ist sie gleich der kanonischen Injektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\hookrightarrow L.}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17e5c440b72cac1651e9298fbfaf133e490646fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.202ex; height:2.176ex;" alt="{\displaystyle K\hookrightarrow L.}" loading="lazy">

Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1},\dotsc ,\omega _{n}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c6ad6c0cb4a686eaf0c995f611d4acf59a677d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.343ex; height:2.009ex;" alt="{\displaystyle \omega _{1},\dotsc ,\omega _{n}}" loading="lazy"> eine Basis der Körpererweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K.}">

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<annotation encoding="application/x-tex">{\displaystyle L/K.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b7f0ae10b6cf4db2fb2a20515de44761f97d50d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.458ex; height:2.843ex;" alt="{\displaystyle L/K.}" loading="lazy"> Also kann jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {A} _{L}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc48ec9480a322d2cdff688e2c97bfca1e41d71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.358ex; height:2.509ex;" alt="{\displaystyle \alpha \in \mathbb {A} _{L}}" loading="lazy"> geschrieben werden als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{j=1}^{n}\alpha _{j}\omega _{j},}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{j=1}^{n}\alpha _{j}\omega _{j},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f1879adbfcc857e1e6423576933c11c73edc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.251ex; height:3.509ex;" alt="{\displaystyle \textstyle \sum _{j=1}^{n}\alpha _{j}\omega _{j},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{j}\in \mathbb {A} _{K}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{j}\in \mathbb {A} _{K}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17b972915089efed757c3459382f15107b20425d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.609ex; height:2.843ex;" alt="{\displaystyle \alpha _{j}\in \mathbb {A} _{K}}" loading="lazy"> eindeutig sind. Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \mapsto \alpha _{j}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a61c49e007868ef4f3a7f0a500964fd70880660a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.499ex; height:2.509ex;" alt="{\displaystyle \alpha \mapsto \alpha _{j}}" loading="lazy"> ist stetig. Definiere nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i,j}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa1408bdf55ca1b365be47447a5aa2d705b65776.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.422ex; height:2.343ex;" alt="{\displaystyle \alpha _{i,j}}" loading="lazy"> (hängen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> ab) via der Gleichungen

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<annotation encoding="application/x-tex">{\displaystyle \alpha \omega _{i}=\sum _{j=1}^{n}\alpha _{i,j}\omega _{j}\qquad \forall i.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/251384aeda225008a653e4d03112bd80eefbb1b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.738ex; height:7.176ex;" alt="{\displaystyle \alpha \omega _{i}=\sum _{j=1}^{n}\alpha _{i,j}\omega _{j}\qquad \forall i.}" loading="lazy"></dd></dl>

Norm und Spur von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> werden definiert als:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&:=\operatorname {Tr} ((\alpha _{i,j})_{i,j})=\sum _{i=1}^{n}\alpha _{i,i}\qquad {\text{ und }}\\N_{L/K}(\alpha )&:=N((\alpha _{i,j})_{i,j})=\operatorname {det} ((\alpha _{i,j})_{i,j}).\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f32cd2e709a00581566cbb9baee007f26c3e985c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:45.419ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&:=\operatorname {Tr} ((\alpha _{i,j})_{i,j})=\sum _{i=1}^{n}\alpha _{i,i}\qquad {\text{ und }}\\N_{L/K}(\alpha )&:=N((\alpha _{i,j})_{i,j})=\operatorname {det} ((\alpha _{i,j})_{i,j}).\end{aligned}}}" loading="lazy"></dd></dl>

Dies sind genau die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> bzw. die <a href="Determinante" title="Determinante">Determinante</a> der <a href="Lineare_Abbildung" title="Lineare Abbildung">linearen Abbildung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{L}\rightarrow \mathbb {A} _{L},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{L}\rightarrow \mathbb {A} _{L},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19f1daa6dc3e480764c1d34448227b63fc1cd8e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.32ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{L}\rightarrow \mathbb {A} _{L},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \alpha x.}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto \alpha x.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56a80bde3a0d0464cb9204875798fa52472c5832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.408ex; height:1.843ex;" alt="{\displaystyle x\mapsto \alpha x.}" loading="lazy"> Beides sind stetige Funktionen auf dem Adelering.

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_von_Norm_und_Spur">Eigenschaften von Norm und Spur</h3></div>

Die Norm und die Spur erfüllen die üblichen Eigenschaften:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha +\beta )&=\operatorname {Tr} _{L/K}(\alpha )+\operatorname {Tr} _{L/K}(\beta )\qquad \forall \alpha ,\beta \in \mathbb {A} _{L},\\\operatorname {Tr} _{L/K}(\operatorname {con} (\alpha ))&=n\alpha \qquad \qquad \qquad \qquad \qquad \forall \alpha \in \mathbb {A} _{K},\\N_{L/K}(\alpha \cdot \beta )&=N_{L/K}(\alpha )\cdot N_{L/K}(\beta )\qquad \forall \alpha ,\beta \in \mathbb {A} _{L},\\N_{L/K}(\operatorname {con} (\alpha ))&=\alpha ^{n}\qquad \qquad \qquad \qquad \qquad \forall \alpha \in \mathbb {A} _{K}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c69b0bcd838b355125ccdb132bae3e8287af7ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:56.702ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha +\beta )&=\operatorname {Tr} _{L/K}(\alpha )+\operatorname {Tr} _{L/K}(\beta )\qquad \forall \alpha ,\beta \in \mathbb {A} _{L},\\\operatorname {Tr} _{L/K}(\operatorname {con} (\alpha ))&=n\alpha \qquad \qquad \qquad \qquad \qquad \forall \alpha \in \mathbb {A} _{K},\\N_{L/K}(\alpha \cdot \beta )&=N_{L/K}(\alpha )\cdot N_{L/K}(\beta )\qquad \forall \alpha ,\beta \in \mathbb {A} _{L},\\N_{L/K}(\operatorname {con} (\alpha ))&=\alpha ^{n}\qquad \qquad \qquad \qquad \qquad \forall \alpha \in \mathbb {A} _{K}.\end{aligned}}}" loading="lazy"></dd></dl>

Weiterhin gilt, dass für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in L}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbef4ff63e30cd41cf6c012293e4c258b31801c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle \alpha \in L}" loading="lazy"> die Spur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} _{L/K}(\alpha )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99ad4e74bf33325c2a3f24fe3f491879e8f319fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.521ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} _{L/K}(\alpha )}" loading="lazy"> und die Norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{L/K}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle N_{L/K}(\alpha )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a563b32e4220b28d66acea523a1fa7e316bacf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.797ex; height:3.176ex;" alt="{\displaystyle N_{L/K}(\alpha )}" loading="lazy"> der üblichen Spur und Norm der Körpererweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

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<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> entspricht. Für einen Körperturm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M/L/K}">

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<annotation encoding="application/x-tex">{\displaystyle M/L/K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f471b9ce64aa342188c5748f9ae970527384c3b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.416ex; height:2.843ex;" alt="{\displaystyle M/L/K}" loading="lazy"> haben wir wie gewohnt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\operatorname {Tr} _{M/L}(\alpha ))&=\operatorname {Tr} _{M/K}(\alpha )\qquad \forall \alpha \in \mathbb {A} _{M},\\N_{L/K}(N_{M/L}(\alpha ))&=N_{M/K}(\alpha )\qquad \forall \alpha \in \mathbb {A} _{M}.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbce98026253c190065dae29521957b83117caf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.349ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\operatorname {Tr} _{M/L}(\alpha ))&=\operatorname {Tr} _{M/K}(\alpha )\qquad \forall \alpha \in \mathbb {A} _{M},\\N_{L/K}(N_{M/L}(\alpha ))&=N_{M/K}(\alpha )\qquad \forall \alpha \in \mathbb {A} _{M}.\end{aligned}}}" loading="lazy"></dd></dl>

Weiterhin kann gezeigt werden:

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</mrow>

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</mtd>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&=\left(\sum _{w\mid v}\operatorname {Tr} _{L_{w}/K_{v}}(\alpha _{w})\right)_{v}\qquad \forall \alpha \in \mathbb {A} _{L}\\N_{L/K}(\alpha )&=\left(\prod _{w\mid v}N_{L_{w}/K_{v}}(\alpha _{w})\right)_{v}\qquad \forall \alpha \in \mathbb {A} _{L}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8e42ecab15a1fe96babe763a9e4c22ff406e684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:48.3ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&=\left(\sum _{w\mid v}\operatorname {Tr} _{L_{w}/K_{v}}(\alpha _{w})\right)_{v}\qquad \forall \alpha \in \mathbb {A} _{L}\\N_{L/K}(\alpha )&=\left(\prod _{w\mid v}N_{L_{w}/K_{v}}(\alpha _{w})\right)_{v}\qquad \forall \alpha \in \mathbb {A} _{L}\end{aligned}}}" loading="lazy"></dd></dl>

Anmerkung: Der letzte Punkt ist nicht trivial, vgl. hierzu <a href="#Literatur">Weil (1967)</a>, S. 52ff bzw. S. 64 oder <a href="#Literatur">Cassels (1967)</a>, S. 74.

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_des_Adelerings">Eigenschaften des Adelerings</h2></div>

Prinzipiell gilt, dass in den Beweisen die Situation oft auf den Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0854290031d92bf519f3ed79754edbe3eec1db77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.973ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} }" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {F} _{p}(t)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {F} _{p}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b43d4fdfdac544970dd486957c1b68b7dc0a8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.293ex; height:3.009ex;" alt="{\displaystyle K=\mathbb {F} _{p}(t)}" loading="lazy"> zurückgeführt werden können. Die Verallgemeinerung für beliebige globale Körper oder ähnliche Objekte ist dann oft trivial.

<div class="mw-heading mw-heading3"><h3 id="Der_Adelering_ist_ein_lokalkompakter,_topologischer_Ring">Der Adelering ist ein lokalkompakter, topologischer Ring</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein <a href="Globaler_K%C3%B6rper" title="Globaler Körper">globaler Körper</a>. Dann ist für jede Stellenmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> der Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{A,S}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{A,S}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cde1d13a054a2f1217a2546e588b4e086b6bdb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.66ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{A,S}}" loading="lazy"> ein <a href="Topologischer_Ring" title="Topologischer Ring">topologischer Ring</a>. Weiterhin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {A} _{A,S},+)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {A} _{A,S},+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/749475a21a9e0896588a44cd8670eeb27ffdd9c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.312ex; height:3.009ex;" alt="{\displaystyle (\mathbb {A} _{A,S},+)}" loading="lazy"> eine <a href="Lokalkompakte_Gruppe" title="Lokalkompakte Gruppe">lokalkompakte Gruppe</a>. Das bedeutet, dass die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K,S}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K,S}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb817225bc479935cafc394e3b9d84c04bf5de21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.889ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K,S}}" loading="lazy"> mit ihrer Topologie lokalkompakt ist und die Gruppenverknüpfung <a href="Stetige_Abbildung" class="mw-redirect" title="Stetige Abbildung">stetig</a> ist. Dies wiederum bedeutet, dass die Abbildung

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}+:\mathbb {A} _{K}\times \mathbb {A} _{K}&\rightarrow \mathbb {A} _{K},\\(a,b)&\mapsto a+b\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/974d1542bc1db43c01d73ef968ca5c54fdbb949c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.761ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}+:\mathbb {A} _{K}\times \mathbb {A} _{K}&\rightarrow \mathbb {A} _{K},\\(a,b)&\mapsto a+b\end{aligned}}}" loading="lazy"></dd></dl>

stetig ist. Darüber hinaus soll auch die Inversionsabbildung der Gruppenverknüpfung stetig sein, d. h. die Abbildung

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}i:\mathbb {A} _{K}\rightarrow \mathbb {A} _{K},\\a&\mapsto -a\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bc1a975afb47a93a4fbf75cc9e2ec4b3a9bd747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.146ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}i:\mathbb {A} _{K}\rightarrow \mathbb {A} _{K},\\a&\mapsto -a\end{aligned}}}" loading="lazy"></dd></dl>

soll stetig sein.

Eine <a href="Umgebungsbasis" title="Umgebungsbasis">Umgebungsbasis</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}(P_{\infty })}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}(P_{\infty })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a07f217a264cdfc534ce9284b89c9433f97ccb4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.548ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}(P_{\infty })}" loading="lazy"> ist auch eine Umgebungsbasis der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> im Adelering. Alternativ bilden auch alle Mengen der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{v}U_{v},}">

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</munder>

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{v}U_{v},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/875ae2bf623fb64f9e31198f6de1d18d63a9de5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.874ex; height:3.009ex;" alt="{\displaystyle \textstyle \prod _{v}U_{v},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{v}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle U_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d9dc89a27eceb76c825ea286a7423f0bc1ef3c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.617ex; height:2.509ex;" alt="{\displaystyle U_{v}}" loading="lazy"> Umgebung der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{v}={\mathcal {O}}_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{v}={\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/930b66b10cdf20ccad55dd0f1379e46205e7d97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.595ex; height:2.509ex;" alt="{\displaystyle U_{v}={\mathcal {O}}_{v}}" loading="lazy"> für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7ba401aee0189f8031d21020a0c640a03339c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.774ex; height:2.009ex;" alt="{\displaystyle v,}" loading="lazy"> eine Umgebungsbasis der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> im Adelering.

Beweisidee: Die Lokalkompaktheit der Menge folgt aus der Definition der restringierten Produkttopologie und der Kompaktheit der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{v}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d6f594d1b3f1caf191e2bc9842685a7e686ed34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{v}.}" loading="lazy"> Die Stetigkeit der Gruppenoperationen lässt sich auf die Stetigkeit der Gruppenoperation in den einzelnen Komponenten zurückführen. Dort sind die entsprechenden Abbildungen stetig. Ein ausführlicherer Beweis findet sich in <a href="#Literatur">Deitmar (2010)</a>, S. 124, Satz 5.2.1.

Bemerkung: Dieses Ergebnis lässt sich auf den Adelering eines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Vektorraums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> und den Adelering einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> übertragen.

<div class="mw-heading mw-heading3"><h3 id="Der_globale_Körper_ist_eine_diskrete,_kokompakte_Untergruppe_in_seinem_Adelering">Der globale Körper ist eine diskrete, kokompakte Untergruppe in seinem Adelering</h3></div>

Der Adelering enthält den globalen Körper als <a href="Diskrete_Untergruppe" title="Diskrete Untergruppe">diskrete</a>, kokompakte Untergruppe, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset \mathbb {A} _{K}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle K\subset \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/035637aa0de3a7578f8fcc6a8126a8237162455e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.536ex; height:2.509ex;" alt="{\displaystyle K\subset \mathbb {A} _{K}}" loading="lazy"> ist diskret und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}/K}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b760420ecdcfbfc87b28bc027103826c7c71283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.6ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}/K}" loading="lazy"> ist in der <a href="Quotiententopologie" title="Quotiententopologie">Quotiententopologie</a> <a href="Kompakter_Raum" title="Kompakter Raum">kompakt</a>. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> <a href="Abgeschlossene_Teilmenge" class="mw-redirect" title="Abgeschlossene Teilmenge">abgeschlossen</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b21c53be4f5448918d406551bf1500d3836e5c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}.}" loading="lazy">

Beweis: Ein Beweis findet sich in <a href="#Literatur">Cassels (1967)</a>, S. 64, Theorem oder in <a href="#Literatur">Weil (1967)</a>, S. 64, Theorem 2. Im Folgenden wird der Beweis für den Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0854290031d92bf519f3ed79754edbe3eec1db77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.973ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} }" loading="lazy"> wiedergegeben:

Um zu zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> diskret in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }}">

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</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af9d3468eef6e7ce070ec0c3125ba4bf03ad145d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.189ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }}" loading="lazy"> ist, reicht es zu zeigen, dass es eine Umgebung der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> gibt, welche keine weiteren Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> enthält. Durch Translation dieser Umgebung kann der allgemeine Fall gezeigt werden. Sei nun

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle U:=\{(\alpha _{p})_{p}:|\alpha _{p}|_{p}\leq 1\quad \forall \,p{\text{ und }}|\alpha _{\infty }|_{\infty }<1\}=\prod \limits _{p<\infty }\mathbb {Z} _{p}\times (-1,1).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7a6d11b1e10689b1294add6ff31d700ea279783.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.676ex; height:5.676ex;" alt="{\displaystyle U:=\{(\alpha _{p})_{p}:|\alpha _{p}|_{p}\leq 1\quad \forall \,p{\text{ und }}|\alpha _{\infty }|_{\infty }<1\}=\prod \limits _{p<\infty }\mathbb {Z} _{p}\times (-1,1).}" loading="lazy"></dd></dl>

Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> eine <a href="Offene_Umgebung" class="mw-redirect" title="Offene Umgebung">offene Umgebung</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }.}">

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</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd6fff3543a0e1e541ce3aa9924de11265cfb018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.836ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }.}" loading="lazy"> Es bleibt zu zeigen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap \mathbb {Q} =\{0\}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\cap \mathbb {Q} =\{0\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58167001c8946b3151d11b4d1ee8a1f0db9028fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.406ex; height:2.843ex;" alt="{\displaystyle U\cap \mathbb {Q} =\{0\}.}" loading="lazy"> Sei dazu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in U\cap \mathbb {Q} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \in U\cap \mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/194ea6419738cc5f29b6150b46d93691a5a52010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.993ex; height:2.509ex;" alt="{\displaystyle \beta \in U\cap \mathbb {Q} .}" loading="lazy"> Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \in \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1a0ec45121c6db7fbff5b95e7e48a6abd953934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.981ex; height:2.509ex;" alt="{\displaystyle \beta \in \mathbb {Q} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\beta |_{p}\leq 1}">

</mrow>

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</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\beta |_{p}\leq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e959a754f0a699db755be365b6744d8d2d7d601.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.946ex; height:3.176ex;" alt="{\displaystyle |\beta |_{p}\leq 1}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> ist, folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in \mathbb {Z} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \in \mathbb {Z} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2443adcf2d882570d7bfa1bdf90ed2fd79024b76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.37ex; height:2.509ex;" alt="{\displaystyle \beta \in \mathbb {Z} .}" loading="lazy"> Da zusätzlich noch gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in (-1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \in (-1,1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a960b38a4d1210204d949b22aae1082cc70d5ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.149ex; height:2.843ex;" alt="{\displaystyle \beta \in (-1,1)}" loading="lazy"> ist, folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22d679b5918dd005eee50595a38e04c35a41eaf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.24ex; height:2.509ex;" alt="{\displaystyle \beta =0.}" loading="lazy">

Nun zur Kompaktheitsaussage: Definiere die Menge

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<annotation encoding="application/x-tex">{\displaystyle W:=\{(\alpha _{p})_{p}:|\alpha _{p}|_{p}\leq 1\quad \forall \,p{\text{ und }}|\alpha _{\infty }|_{\infty }\leq 1/2\}=\prod \limits _{p<\infty }\mathbb {Z} _{p}\times [-{\frac {1}{2}},{\frac {1}{2}}].}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9136fb255f46fba66d2549282b7b950ac619b4fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:69.811ex; height:6.509ex;" alt="{\displaystyle W:=\{(\alpha _{p})_{p}:|\alpha _{p}|_{p}\leq 1\quad \forall \,p{\text{ und }}|\alpha _{\infty }|_{\infty }\leq 1/2\}=\prod \limits _{p<\infty }\mathbb {Z} _{p}\times [-{\frac {1}{2}},{\frac {1}{2}}].}" loading="lazy"></dd></dl>

Wir zeigen nun, dass jede <a href="%C3%84quivalenzklasse" class="mw-redirect" title="Äquivalenzklasse">Klasse</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }">

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</mrow>

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</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/372a8148fb0f72f9996ca83a35aa7884acfbdb22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.159ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }" loading="lazy"> einen Vertreter in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> hat, das heißt wir müssen zeigen, dass für jedes Adel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {A} _{\mathbb {Q} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {A} _{\mathbb {Q} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c257d08e99166ad1e7997966cb2ca4c91994b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.517ex; height:2.676ex;" alt="{\displaystyle \alpha \in \mathbb {A} _{\mathbb {Q} }}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \in \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \in \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1a0ec45121c6db7fbff5b95e7e48a6abd953934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.981ex; height:2.509ex;" alt="{\displaystyle \beta \in \mathbb {Q} }" loading="lazy"> existiert, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha -\beta \in W.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha -\beta \in W.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94049dd8b225fbf10936d3e1c8d4604f591ffcb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.583ex; height:2.509ex;" alt="{\displaystyle \alpha -\beta \in W.}" loading="lazy"> Sei nun also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =(\alpha _{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =(\alpha _{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/342d52325d23d8f6c151fbb84df723053546f329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.031ex; height:3.009ex;" alt="{\displaystyle \alpha =(\alpha _{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}" loading="lazy"> beliebig. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> eine <a href="Primzahl" title="Primzahl">Primzahl</a> für die gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha _{p}|>1.}">

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\alpha _{p}|>1.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/908aea0578d0234eadecbc80e0f884a80fc4041d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.748ex; height:3.009ex;" alt="{\displaystyle |\alpha _{p}|>1.}" loading="lazy"> Dann existiert ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{p}=z_{p}/p^{x_{p}}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{p}=z_{p}/p^{x_{p}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bf7143bcc46fdd56db26ca55795b9917ff07820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.687ex; height:3.009ex;" alt="{\displaystyle r_{p}=z_{p}/p^{x_{p}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{p}\in \mathbb {Z} ,}">

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<annotation encoding="application/x-tex">{\displaystyle z_{p}\in \mathbb {Z} ,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3674855fe78164f39d2da78991f0b7422e90df6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.178ex; height:2.843ex;" alt="{\displaystyle z_{p}\in \mathbb {Z} ,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle x_{p}\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f3b111f279413d444e251ae58e13aba3a0e3ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.908ex; height:2.843ex;" alt="{\displaystyle x_{p}\in \mathbb {N} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha _{p}-r_{p}|\leq 1.}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle |\alpha _{p}-r_{p}|\leq 1.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0003fcc0b643d26886d904cc7ef61220742b75d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.696ex; height:3.009ex;" alt="{\displaystyle |\alpha _{p}-r_{p}|\leq 1.}" loading="lazy"> Nun ersetze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha -r_{p}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha -r_{p}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e10b128942e01f48bebdc1eeb0df7330598dd7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.083ex; height:2.676ex;" alt="{\displaystyle \alpha -r_{p}.}" loading="lazy"> Dies beeinflusst die anderen Stellen wie folgt:

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\neq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\neq p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b15493bd4b96c9d3ccf0a3e9ca7e8b12b5ae99a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.337ex; height:2.676ex;" alt="{\displaystyle q\neq p}" loading="lazy"> eine weitere Primzahl. Dann gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq \operatorname {max} \{|a_{q}|_{q},|r_{p}|_{q}\}\leq \operatorname {max} \{|a_{q}|_{q},1\}\leq 1.}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq \operatorname {max} \{|a_{q}|_{q},|r_{p}|_{q}\}\leq \operatorname {max} \{|a_{q}|_{q},1\}\leq 1.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88dce6bf43f6d90c2daa3cac2b37164634c9e9ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:50.734ex; height:3.176ex;" alt="{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq \operatorname {max} \{|a_{q}|_{q},|r_{p}|_{q}\}\leq \operatorname {max} \{|a_{q}|_{q},1\}\leq 1.}" loading="lazy"> Es folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq 1\Leftrightarrow |\alpha _{q}|_{q}\leq 1}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq 1\Leftrightarrow |\alpha _{q}|_{q}\leq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45be290d9e2cf68ef6d993f3c59f389bd65c5ba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.601ex; height:3.176ex;" alt="{\displaystyle |\alpha _{q}-r_{p}|_{q}\leq 1\Leftrightarrow |\alpha _{q}|_{q}\leq 1}" loading="lazy"> (für die Hinrichtung ist zu beachten, dass in der scharfen Dreiecksungleichung Gleichheit gilt, falls die Beträge der beiden beteiligten Zahlen verschieden sind).

Damit haben wir die (endliche) Primstellenmenge mit der Eigenschaft, dass die Komponenten nicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy"> liegen, um eins verkleinert. Iteration liefert die Existenz eines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in \mathbb {Q} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {Q} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce2d8c1a781840c28155b91f54b6f4aab1047676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.344ex; height:2.509ex;" alt="{\displaystyle r\in \mathbb {Q} ,}" loading="lazy"> sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha -r\in {\widehat {\mathbb {Z} }}\times \mathbb {R} }">

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha -r\in {\widehat {\mathbb {Z} }}\times \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6946af0a224224290dd96c33156ed05d7b8feb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.286ex; height:3.009ex;" alt="{\displaystyle \alpha -r\in {\widehat {\mathbb {Z} }}\times \mathbb {R} }" loading="lazy"> ist. Jetzt wähle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {Z} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {Z} ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38866e48a32b7877310c4a577e56072086260958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.128ex; height:2.509ex;" alt="{\displaystyle s\in \mathbb {Z} ,}" loading="lazy"> so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\infty }-r-s\in [-1/2,1/2].}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{\infty }-r-s\in [-1/2,1/2].}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a46d2085dd527bfffb217ac52ad7f6b7708c1cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.781ex; height:2.843ex;" alt="{\displaystyle \alpha _{\infty }-r-s\in [-1/2,1/2].}" loading="lazy"> Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65926436c4515b09d7989e4c528e7c9839b9bcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.481ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {Z} }" loading="lazy"> folgt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha -\beta \in W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha -\beta \in W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1a50842e01ddad9e9d01fd9669c8bb1db9a70f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.936ex; height:2.509ex;" alt="{\displaystyle \alpha -\beta \in W}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta :=r+s\in \mathbb {Q} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta :=r+s\in \mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/169cbe25a107a37eebc651f77addd325be8c08f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.352ex; height:2.509ex;" alt="{\displaystyle \beta :=r+s\in \mathbb {Q} .}" loading="lazy"> Betrachte nun die stetige Projektion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :W\twoheadrightarrow \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} .}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :W\twoheadrightarrow \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b03f8a5bde4aec6e04ac8c067dfedfb933d3dcc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.125ex; height:2.843ex;" alt="{\displaystyle \pi :W\twoheadrightarrow \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} .}" loading="lazy"> Sie ist surjektiv. Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/372a8148fb0f72f9996ca83a35aa7884acfbdb22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.159ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }" loading="lazy"> das stetige Bild eines Kompaktums, also selbst kompakt. Der Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {F} _{p}(t)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {F} _{p}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b43d4fdfdac544970dd486957c1b68b7dc0a8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.293ex; height:3.009ex;" alt="{\displaystyle K=\mathbb {F} _{p}(t)}" loading="lazy"> geht ähnlich.

Der Zusatz ist ein Lemma über <a href="Topologische_Gruppe" title="Topologische Gruppe">topologische Gruppen</a>.

Korollar: Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ein endlichdimensionaler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-<a href="Vektorraum" title="Vektorraum">Vektorraum</a>. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> diskret in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> und kokompakt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E},}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/424a6cffcc905ca58a1ffc3e8743d66467b68780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.813ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E},}" loading="lazy"> d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}/E}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}/E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/840a439f4696188b30875c00271f67ea7367a120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.104ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{E}/E}" loading="lazy"> ist kompakt.

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_des_rationalen_Adelerings">Eigenschaften des rationalen Adelerings</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Z} }={\widehat {\mathbb {Z} }}\times \mathbb {R} =\prod _{p}\mathbb {Z} _{p}\times \mathbb {R} }">

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Z} }={\widehat {\mathbb {Z} }}\times \mathbb {R} =\prod _{p}\mathbb {Z} _{p}\times \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbf742179cba562ac418610e385d2cb5b13c091e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.757ex; height:5.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Z} }={\widehat {\mathbb {Z} }}\times \mathbb {R} =\prod _{p}\mathbb {Z} _{p}\times \mathbb {R} }" loading="lazy"> wie zuvor. Dann gilt:

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {A} _{\mathbb {Q} }&=\mathbb {Q} +\mathbb {A} _{\mathbb {Z} }{\text{ und }}\\\mathbb {Z} &=\mathbb {Q} \cap \mathbb {A} _{\mathbb {Z} }.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b80e55c9a1ce4c7b067e2bc8f7e08e9cb69cc5fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.733ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbb {A} _{\mathbb {Q} }&=\mathbb {Q} +\mathbb {A} _{\mathbb {Z} }{\text{ und }}\\\mathbb {Z} &=\mathbb {Q} \cap \mathbb {A} _{\mathbb {Z} }.\end{aligned}}}" loading="lazy"></dd></dl>

Weiterhin gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba6e32204c68d701afce63d5769c0c7c6add6619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.902ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> uneingeschränkt

divisibel ist, d. h. die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nx=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e7accc191f4cefe060c28bfaec2f95c7a3c5094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.978ex; height:2.009ex;" alt="{\displaystyle nx=y}" loading="lazy"> hat für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e1d6eb156ffa4502875897a26689e199f1b6c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.898ex; height:2.843ex;" alt="{\displaystyle y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> eine

Lösung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d07e9f568a88785ae48006ac3c4b951020f1699a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}" loading="lazy"> Allerdings ist diese Lösung im Allgemeinen nicht eindeutig.

Außerdem gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} ,fin}}">

</mrow>

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} ,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1affe5973050aea3a64ad1438e3ce22420d7e3b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.011ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} ,fin}}" loading="lazy"> dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}">

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f68751aa23c5257ad5f77a307adbc5feaacce4db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.104ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}" loading="lazy"> ist. Eine allgemeinere Formulierung dieser Aussage findet sich im <a href="#„Approximation_Theorems“">Satz über starke Approximation</a>.

Beweis: Die ersten Aussagen können elementar bewiesen werden. Die nächste Aussage findet sich so in <a href="#Literatur">Neukirch (2007)</a> auf Seite 383. Wir beweisen sie im Folgenden. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e1d6eb156ffa4502875897a26689e199f1b6c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.898ex; height:2.843ex;" alt="{\displaystyle y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> beliebig. Zu zeigen: Es existiert ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fbab77c9cb7f85666ea9dde7e9281fcb548f066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.072ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> sodass gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx=y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nx=y.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3de3942253bcae3d7d1d4bda45a695dc1ad186b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.625ex; height:2.009ex;" alt="{\displaystyle nx=y.}" loading="lazy"> Wir zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af9d3468eef6e7ce070ec0c3125ba4bf03ad145d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.189ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }}" loading="lazy"> uneingeschränkt reversibel ist, dann folgt bereits die Behauptung. Dies ist jedoch klar, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af9d3468eef6e7ce070ec0c3125ba4bf03ad145d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.189ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }}" loading="lazy"> in jeder Koordinate ein Körper mit Charakteristik ungleich Null ist. Nun zu einem Gegenbeispiel, welches zeigt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba6e32204c68d701afce63d5769c0c7c6add6619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.902ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> nicht eindeutig reversibel ist. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=(0,0,\dotsc ,0)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=(0,0,\dotsc ,0)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/386398e6e1c85d9b834170d2791b88539e4d94ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.896ex; height:2.843ex;" alt="{\displaystyle y=(0,0,\dotsc ,0)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6bf67f9d06ca3af619657f8d20ee1322da77174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}" loading="lazy"> beliebig. Dann erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=(0,0,\dotsc ,0)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}=(0,0,\dotsc ,0)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cc329431c11ed7816695d7742894fcb85691805.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.124ex; height:2.843ex;" alt="{\displaystyle x_{1}=(0,0,\dotsc ,0)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx=y.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nx=y.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3de3942253bcae3d7d1d4bda45a695dc1ad186b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.625ex; height:2.009ex;" alt="{\displaystyle nx=y.}" loading="lazy"> Ebenfalls erfüllt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=(1/n,1/n,\dotsc ,1/n)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}=(1/n,1/n,\dotsc ,1/n)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acfba1a82c7d33a83a22fd70b36d658d34d0dc79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.796ex; height:2.843ex;" alt="{\displaystyle x_{2}=(1/n,1/n,\dotsc ,1/n)+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} }" loading="lazy"> diese Gleichungen, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx=(1,1,\dotsc ,1)+\mathbb {Z} =y.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nx=(1,1,\dotsc ,1)+\mathbb {Z} =y.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/118da7c244af6961e11242a49dbd248f943d421f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.623ex; height:2.843ex;" alt="{\displaystyle nx=(1,1,\dotsc ,1)+\mathbb {Z} =y.}" loading="lazy"> Da n nur endlich viele Teiler hat, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy"> wohldefiniert. Aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\neq x_{2},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}\neq x_{2},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d77d7f03fbe8d415057b67420ecb4b180ddc0b5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.513ex; height:2.676ex;" alt="{\displaystyle x_{1}\neq x_{2},}" loading="lazy"> denn (betrachte unendliche Koordinate) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n\neq 0+z\,\,\forall z\in \mathbb {Z} .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/n\neq 0+z\,\,\forall z\in \mathbb {Z} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/675a92cdfbd4bf360b5137aa93dead5deac04c90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.102ex; height:2.843ex;" alt="{\displaystyle 1/n\neq 0+z\,\,\forall z\in \mathbb {Z} .}" loading="lazy">

Bemerkung: In unserem Fall ist die eindeutige Reversibilität äquivalent zur <a href="Torsionsfreiheit" class="mw-redirect" title="Torsionsfreiheit">Torsionsfreiheit</a> und die ist hier nicht gegeben, da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot x_{2}=0,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\cdot x_{2}=0,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff3732bbc5ab06611780239ba8ae657298ca589e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.365ex; height:2.509ex;" alt="{\displaystyle n\cdot x_{2}=0,}" loading="lazy"> aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}\neq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0e0b0b810812acd81e12743fc8b070f7a641c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x_{2}\neq 0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\neq 0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\neq 0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6ca8d061c2c701f5f4ed0a5b805f316e17c24a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.302ex; height:2.676ex;" alt="{\displaystyle n\neq 0.}" loading="lazy">

Zur letzten Aussage: Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}=\mathbb {Q} {\widehat {\mathbb {Z} }},}">

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}=\mathbb {Q} {\widehat {\mathbb {Z} }},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee52cd477c67c369d2b7537775f298ea017f789a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.208ex; height:3.509ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}=\mathbb {Q} {\widehat {\mathbb {Z} }},}" loading="lazy"> da wir die endlich vielen Nenner in den Koordinaten der Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}">

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f68751aa23c5257ad5f77a307adbc5feaacce4db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.104ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}" loading="lazy"> durch ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\in \mathbb {Q} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03aaa9327494402fb81829278f439afcf201313a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.718ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {Q} }" loading="lazy"> erreichen können. Wenn wir zeigen können, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbb {Z} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12112fb1afd12580b51b080bb33356edd1d0c797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }}}" loading="lazy"> ist, folgt dann bereits die Behauptung. Es ist also zu zeigen, dass sich in jeder offenen Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbb {Z} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12112fb1afd12580b51b080bb33356edd1d0c797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }}}" loading="lazy"> ein Element aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> befindet. Die offene Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> kann ohne Einschränkung als

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\prod _{p\in E}\left(a_{p}+p^{l_{p}}\mathbb {Z} _{p}\right)\times \prod _{p\notin E}\mathbb {Z} _{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f60fa5817288e4de219da82845a205ab3b7c7db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.063ex; height:6.009ex;" alt="{\displaystyle V=\prod _{p\in E}\left(a_{p}+p^{l_{p}}\mathbb {Z} _{p}\right)\times \prod _{p\notin E}\mathbb {Z} _{p}}" loading="lazy"></dd></dl>

angenommen werden, denn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p^{m}\mathbb {Z} _{p})_{m\in \mathbb {N} }}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p^{m}\mathbb {Z} _{p})_{m\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1fbd9a14e9f71e8cebcc36d25c40a345195339a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.221ex; height:3.009ex;" alt="{\displaystyle (p^{m}\mathbb {Z} _{p})_{m\in \mathbb {N} }}" loading="lazy"> bilden eine Umgebungsbasis der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2d1fd592239d60f9a9f36feb4b3be40df666f6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}.}" loading="lazy">

Mit Hilfe des <a href="Chinesischer_Restsatz" title="Chinesischer Restsatz">Chinesischen Restsatzes</a> zeigt man nun die Existenz eines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb0c3677d7d07beedd44861759004f5827ab99cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.084ex; height:2.176ex;" alt="{\displaystyle l\in \mathbb {Z} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\equiv a_{p}\;{\bmod {\;}}p^{l_{p}},}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\equiv a_{p}\;{\bmod {\;}}p^{l_{p}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f335d521cc8fb5a835af94dc7cb3d5a1bb2af6a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.427ex; height:3.343ex;" alt="{\displaystyle l\equiv a_{p}\;{\bmod {\;}}p^{l_{p}},}" loading="lazy"> da Primzahlpotenzen zu verschiedenen Primzahlen teilerfremd sind. Dies bedeutet so viel wie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\in V.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\in V.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6abd73ba1f67c611a65a5f4328ee4bb9ba97b0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.968ex; height:2.176ex;" alt="{\displaystyle l\in V.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Haarmaß_auf_dem_Adelering">Haarmaß auf dem Adelering</h3></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {A} _{K},+)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {A} _{K},+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f292926b7fc8cadd3e34fbe0e574217799bfa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.023ex; height:2.843ex;" alt="{\displaystyle (\mathbb {A} _{K},+)}" loading="lazy"> eine lokalkompakte Gruppe. Folglich existiert ein <a href="Haarma%C3%9F" class="mw-redirect" title="Haarmaß">Haarmaß</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dx}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/845c817e348381a13f3fad5184169ce0e021c685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}" loading="lazy"> auf dieser Gruppe, welches folgendermaßen normalisiert werden kann:

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> eine einfache Funktion auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K},}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bef6ca8bc0051c81dee17fa2d17e74c6029d22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K},}" loading="lazy"> d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f=\prod _{v}f_{v},}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f=\prod _{v}f_{v},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90d6566f2319e752d19bef362c7a2e91d76c5ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.803ex; height:3.009ex;" alt="{\displaystyle \textstyle f=\prod _{v}f_{v},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{v}:K_{v}\rightarrow \mathbb {C} }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{v}:K_{v}\rightarrow \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82bd586a245ef17002d92a3c3de05f945f414921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.401ex; height:2.509ex;" alt="{\displaystyle f_{v}:K_{v}\rightarrow \mathbb {C} }" loading="lazy"> messbar und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76af6ceb7fac9a8e080a02220ad1208085a0bfcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.956ex; height:2.676ex;" alt="{\displaystyle f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}}" loading="lazy"> für fast alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb7b87f7afb1452b9d0e287f8ef746a9912c8333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle v.}" loading="lazy"> Das Haarmaß <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dx}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/845c817e348381a13f3fad5184169ce0e021c685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> kann so normalisiert werden, dass für jede

integrierbare, einfache Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f=\prod _{v}f_{v}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f=\prod _{v}f_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e889ab4d7cf51711cb17d4e118ecdc5340df443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.156ex; height:3.009ex;" alt="{\displaystyle \textstyle f=\prod _{v}f_{v}}" loading="lazy"> die Produktformel

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {A} }f(x)dx=\prod _{v}\int _{K_{v}}f_{v}dx_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c80480e2f98edb4d28578c90ff77e1264f9fd35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.379ex; height:6.343ex;" alt="{\displaystyle \int _{\mathbb {A} }f(x)dx=\prod _{v}\int _{K_{v}}f_{v}dx_{v}}" loading="lazy"></dd></dl>

gilt, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int _{{\mathcal {O}}_{v}}1dx_{v}=1}">

<msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{{\mathcal {O}}_{v}}1dx_{v}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b244cf50d7669284c293318b0c6179da165022f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.835ex; height:3.509ex;" alt="{\displaystyle \textstyle \int _{{\mathcal {O}}_{v}}1dx_{v}=1}" loading="lazy"> für jede endliche Stelle gilt. An den unendlichen Stellen wird das <a href="Lebesgue-Ma%C3%9F" title="Lebesgue-Maß">Lebesgue-Maß</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"> genommen.

Um einzusehen, warum das Maß so normalisiert werden kann, wird es zuerst auf den sogenannten einfachen Mengen (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{v}A_{v}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{v}A_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa71986b5212ac585ee37378af9f8c633b6932ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.383ex; height:3.009ex;" alt="{\displaystyle \textstyle \prod _{v}A_{v}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{v}\subset K_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{v}\subset K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fb8a28b234be40fe643305858256c5911064859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.874ex; height:2.509ex;" alt="{\displaystyle A_{v}\subset K_{v}}" loading="lazy"> offen und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{v}={\mathcal {O}}_{v}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{v}={\mathcal {O}}_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88ee97f3cf94c6d29f26cf9eac5b5415fd6d9800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.751ex; height:2.509ex;" alt="{\displaystyle A_{v}={\mathcal {O}}_{v}}" loading="lazy"> fast immer) definiert und dann auf die ganze Borel-σ-Algebra fortgesetzt. Dies findet sich in <a href="#Literatur">Deitmar (2010)</a>, S. 126, Satz 5.2.2.

Es kann gezeigt werden, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}/K}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b760420ecdcfbfc87b28bc027103826c7c71283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.6ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{K}/K}" loading="lazy"> endliches Volumen im Quotientenmaß hat. Das Quotientenmaß wird vom Haarmaß auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> induziert. Diese Aussage ist ein Korollar aus dem obigen Satz, da die Kompaktheit das endliche Maß dieser Menge impliziert.

<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Endlichkeit_der_Klassenzahl_eines_Zahlkörpers">Endlichkeit der Klassenzahl eines Zahlkörpers</h3></div>

In diesem Abschnitt wollen wir die Endlichkeit der <a href="Klassenzahl" title="Klassenzahl">Klassenzahl</a> für einen algebraischen Zahlkörper beweisen. Dies kann auf verschiedene Weisen erfolgen. Im Beweis des <a href="Idelegruppe#Charakterisierung_der_Idelegruppe" title="Idelegruppe">Satzes über die Charakterisierung der Idelegruppe</a> wird dieser Endlichkeitssatz schon verwendet. Es gibt im Wesentlichen zwei Herangehensweisen: Im einen Fall zeigt man zuerst die Endlichkeit der Klassenzahl und leitet dann die Resultate über Adele und Idele ab, im anderen Fall folgert man die Endlichkeit der Klassenzahl aus diesen Resultaten.

Satz (Endlichkeit der Klassenzahl eines Zahlkörpers):

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein algebraischer Zahlkörper. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |Cl_{K}|<\infty .}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |Cl_{K}|<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54edd4e175ac1075a14a757eec95c9e755cff5dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.515ex; height:2.843ex;" alt="{\displaystyle |Cl_{K}|<\infty .}" loading="lazy">

Beweis der Endlichkeit der Klassenzahl: Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}^{1}\rightarrow J_{K},((\alpha _{v})_{v<\infty },(\alpha _{v})_{v\mid \infty })\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}^{1}\rightarrow J_{K},((\alpha _{v})_{v<\infty },(\alpha _{v})_{v\mid \infty })\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e1a5a80be127c79718954bd6fa457e8536ea14a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.635ex; height:5.676ex;" alt="{\displaystyle \mathbb {A} _{K}^{1}\rightarrow J_{K},((\alpha _{v})_{v<\infty },(\alpha _{v})_{v\mid \infty })\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}}" loading="lazy"> ist surjektiv und deswegen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Cl_{K}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Cl_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35029619632bbed91a85e9d42009a52ce0743a62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.153ex; height:2.509ex;" alt="{\displaystyle Cl_{K}}" loading="lazy"> das stetige Bild des Kompaktums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}^{1}/K^{\times },}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}^{1}/K^{\times },}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f05b95605bcef378fd828794e112e646608884c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.785ex; height:3.176ex;" alt="{\displaystyle \mathbb {A} _{K}^{1}/K^{\times },}" loading="lazy"> also kompakt. Gleichzeitig ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Cl_{K}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Cl_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35029619632bbed91a85e9d42009a52ce0743a62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.153ex; height:2.509ex;" alt="{\displaystyle Cl_{K}}" loading="lazy"> auch diskret, also endlich.

Bemerkung: Ein ähnliches Ergebnis gilt auch für den Funktionenkörperfall. Hier wird eine sogenannte Divisorgruppe („divisor group“) von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> definiert und man kann zeigen, dass die Divisoren von Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> modulo der Menge der Hauptdivisoren eine endliche Gruppe bilden (dies sind die analogen Begriffe im Funktionenkörperfall) (vgl. <a href="#Literatur">Cassels (1967)</a>, S. 71).

<div class="mw-heading mw-heading3"><h3 id="Einheiten_und_Dirichletscher_Einheitensatz">Einheiten und Dirichletscher Einheitensatz</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Definition_und_Eigenschaften_erster_Hilfsgrößen">Definition und Eigenschaften erster Hilfsgrößen</h4></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> eine endliche Teilmenge der Stellenmenge, welche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> enthält. Definiere

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Omega (P)&:=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}{\mathcal {O}}_{v}^{\times }=(\mathbb {A} _{K}(P))^{\times },\\E(P)&:=K^{\times }\cap \Omega (P).\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5523b3ac547c849b6d2c1a75acb6a736065db378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.692ex; margin-bottom: -0.312ex; width:40.654ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\Omega (P)&:=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}{\mathcal {O}}_{v}^{\times }=(\mathbb {A} _{K}(P))^{\times },\\E(P)&:=K^{\times }\cap \Omega (P).\end{aligned}}}" loading="lazy"></dd></dl>

Dann gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc08a0b9b40bb460b541adc55bc55e08f3378f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle E(P)}" loading="lazy"> eine Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98403ab9e2f27fde4576ca3e622add252f4fd9a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.343ex;" alt="{\displaystyle K^{\times }}" loading="lazy"> ist, welche alle Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in K^{\times }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in K^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89bd56eae98d5fd109d06f578a34a492fc3f1f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.475ex; height:2.676ex;" alt="{\displaystyle \xi \in K^{\times }}" loading="lazy"> enthält, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(\xi )=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(\xi )=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0a12fbcb0d3e629883147eac2d3d3eb714dcbcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.228ex; height:2.843ex;" alt="{\displaystyle v(\xi )=0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\notin P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\notin P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2789df6a6488cd3cbfc76a8f911e332d30e055e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.714ex; height:2.676ex;" alt="{\displaystyle v\notin P}" loading="lazy"> erfüllen. Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98403ab9e2f27fde4576ca3e622add252f4fd9a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.343ex;" alt="{\displaystyle K^{\times }}" loading="lazy"> diskret in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{K}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3887e58b3ee5ff42e9d95ff499ceb85386a2e565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle I_{K}}" loading="lazy"> ist, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc08a0b9b40bb460b541adc55bc55e08f3378f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle E(P)}" loading="lazy"> eine diskrete Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5bbe7288ee842e07f3e359c324ebd90d2d37140.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.233ex; height:2.843ex;" alt="{\displaystyle \Omega (P)}" loading="lazy"> und ebenfalls von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{1}(P):=\Omega (P)\cap \mathbb {A} _{K}^{1}.}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{1}(P):=\Omega (P)\cap \mathbb {A} _{K}^{1}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc254d88876649619743dd0fe6a113f097aed68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.866ex; height:3.176ex;" alt="{\displaystyle \Omega _{1}(P):=\Omega (P)\cap \mathbb {A} _{K}^{1}.}" loading="lazy">

Eine alternative Definition von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc08a0b9b40bb460b541adc55bc55e08f3378f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle E(P)}" loading="lazy"> ist, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(P)=K(P)^{\times },}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(P)=K(P)^{\times },}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6733569e69eab15dda5bc5830d90d50a7ddfc4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.207ex; height:2.843ex;" alt="{\displaystyle E(P)=K(P)^{\times },}" loading="lazy"> wobei der Unterring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ad52f4f7bbeef99218890830e6614a1cd998580.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.621ex; height:2.843ex;" alt="{\displaystyle K(P)}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> gegeben ist durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle K(P):=K\cap (\prod _{v\in P}K_{v}\times \mathbb {A} _{K}(P).}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle K(P):=K\cap (\prod _{v\in P}K_{v}\times \mathbb {A} _{K}(P).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d30bb1acabc0332d33f7b9e1259e1cdf622ebb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.276ex; height:3.009ex;" alt="{\displaystyle \textstyle K(P):=K\cap (\prod _{v\in P}K_{v}\times \mathbb {A} _{K}(P).}" loading="lazy"> Also enthält <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ad52f4f7bbeef99218890830e6614a1cd998580.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.621ex; height:2.843ex;" alt="{\displaystyle K(P)}" loading="lazy"> alle Elemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c35dd8426f3bb7e6d89b222bb691acd480b57ae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.584ex; height:2.509ex;" alt="{\displaystyle \xi \in K,}" loading="lazy"> für die gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(\xi )\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(\xi )\geq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4668069db7d49071f0ccf1a304117eab4451ffa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.228ex; height:2.843ex;" alt="{\displaystyle v(\xi )\geq 0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\notin P.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\notin P.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4d6bc8405eb23b94fdb2e4add697edb713c05f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.676ex;" alt="{\displaystyle v\notin P.}" loading="lazy">

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<c<C<\infty .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<c<C<\infty .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f28f300869d29df9d0562f87e31c2937fd9c07b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.202ex; height:2.176ex;" alt="{\displaystyle 0<c<C<\infty .}" loading="lazy"> Dann ist die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\eta \in E(P):|\eta _{v}|_{v}=1\quad \forall v\notin P{\text{ und }}c\leq |\eta _{v}|_{v}\leq C\quad \forall v\in P\}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\eta \in E(P):|\eta _{v}|_{v}=1\quad \forall v\notin P{\text{ und }}c\leq |\eta _{v}|_{v}\leq C\quad \forall v\in P\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4b4b993c1606ebecdee584d3c2994cb6827e5fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.547ex; height:3.009ex;" alt="{\displaystyle \{\eta \in E(P):|\eta _{v}|_{v}=1\quad \forall v\notin P{\text{ und }}c\leq |\eta _{v}|_{v}\leq C\quad \forall v\in P\}}" loading="lazy"> endlich. Um dies einzusehen, definiere

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W:=\{(\alpha _{v})_{v}:|\alpha _{v}|_{v}=1\quad \forall v\notin P{\text{ und }}c\leq |\alpha _{v}|_{v}\leq C\quad \forall v\in P\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7886e6c9a9f2926b5e85eaaad5437357bbd973a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.054ex; height:3.009ex;" alt="{\displaystyle W:=\{(\alpha _{v})_{v}:|\alpha _{v}|_{v}=1\quad \forall v\notin P{\text{ und }}c\leq |\alpha _{v}|_{v}\leq C\quad \forall v\in P\}.}" loading="lazy"></dd></dl>

Dann gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> kompakt ist und die oben beschriebene Menge ist der Schnitt aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> mit der diskreten Untergruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98403ab9e2f27fde4576ca3e622add252f4fd9a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.343ex;" alt="{\displaystyle K^{\times }}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{K}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ee595772bc132bf2d35036e73083f0fd255fe45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.363ex; height:2.509ex;" alt="{\displaystyle I_{K}.}" loading="lazy"> Daraus folgt die Endlichkeit der oben beschriebenen Menge.

Definiere nun weiterhin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:=\{\xi \in K:|\xi |_{v}\leq 1\quad \forall v\}=\{\xi \in K:|\xi |_{v}=1\quad \forall v\}\cup \{0\}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:=\{\xi \in K:|\xi |_{v}\leq 1\quad \forall v\}=\{\xi \in K:|\xi |_{v}=1\quad \forall v\}\cup \{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bedda2218edea3876a0d8f3b429fb3535caaddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.766ex; height:3.009ex;" alt="{\displaystyle F:=\{\xi \in K:|\xi |_{v}\leq 1\quad \forall v\}=\{\xi \in K:|\xi |_{v}=1\quad \forall v\}\cup \{0\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E:=F\setminus \{0\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E:=F\setminus \{0\},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/905d54d186ab506971b6f8e1ebdd810e8cfff907.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.591ex; height:2.843ex;" alt="{\displaystyle E:=F\setminus \{0\},}" loading="lazy"> wobei die Gleichheit auf Grund der allgemeinen Produktformel gilt. Dann gilt

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\subset E(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cad5d9ff6532b9d42de7b7c514b73509d4f27bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.204ex; height:2.843ex;" alt="{\displaystyle E\subset E(P)}" loading="lazy"></dd></dl>

für jede endliche Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> der Stellenmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> welche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbead2b781bd0b4b817a54594bc01ceeb0b5b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{\infty }}" loading="lazy"> enthält.

<div class="mw-heading mw-heading4"><h4 id="Satz:_Einheitswurzeln_von_K">Satz: Einheitswurzeln von K</h4></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ist eine endliche zyklische Gruppe, welche alle Einheitswurzeln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> enthält. Es folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> gerade die Gruppe der Einheitswurzeln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist.

Beweis: Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=K\cap \{(x_{v})_{v}\in \mathbb {A} _{K}:|x_{v}|_{v}\leq 1\quad \forall v\}.}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=K\cap \{(x_{v})_{v}\in \mathbb {A} _{K}:|x_{v}|_{v}\leq 1\quad \forall v\}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8ab5fe451067264f07b23e05848123c0e23f72f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.493ex; height:3.009ex;" alt="{\displaystyle F=K\cap \{(x_{v})_{v}\in \mathbb {A} _{K}:|x_{v}|_{v}\leq 1\quad \forall v\}.}" loading="lazy"> Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> diskret in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> ist, ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> diskret in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}.}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b21c53be4f5448918d406551bf1500d3836e5c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}.}" loading="lazy"> Weiterhin ist die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{v})_{v}\in \mathbb {A} _{K}:|x_{v}|_{v}\leq 1\quad \forall v\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x_{v})_{v}\in \mathbb {A} _{K}:|x_{v}|_{v}\leq 1\quad \forall v\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f74c748025863883cc7e6df92d5c427b727f4d7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.358ex; height:3.009ex;" alt="{\displaystyle \{(x_{v})_{v}\in \mathbb {A} _{K}:|x_{v}|_{v}\leq 1\quad \forall v\}}" loading="lazy"> kompakt und damit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> eine Teilmenge einer kompakten Menge. Es folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> endlich ist. Wegen der allgemeinen Produktformel gilt für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in E,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f06b7c051c5f1e6b14ebda1f1d450efd8cb3ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.293ex; height:2.509ex;" alt="{\displaystyle \xi \in E,}" loading="lazy"> dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\xi |_{v}=1}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\xi |_{v}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a77167114098243f85658752548f55786ce96960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.614ex; height:3.009ex;" alt="{\displaystyle |\xi |_{v}=1}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb7b87f7afb1452b9d0e287f8ef746a9912c8333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:1.676ex;" alt="{\displaystyle v.}" loading="lazy"> Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> eine endliche Untergruppe von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{\times }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c375de480cd169959de091fbd020222eecd74b4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.252ex; height:2.343ex;" alt="{\displaystyle K^{\times }.}" loading="lazy"> Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein Körper ist, folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> zyklisch ist. Offensichtlich liegt jede Einheitswurzel von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89862747e88ca143e979241a9a243b5ef66ddc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.509ex;" alt="{\displaystyle E,}" loading="lazy"> da alle Einheitswurzeln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> Betrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> und damit Bewertung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> haben. Angenommen, es existiert ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in E,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in E,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f06b7c051c5f1e6b14ebda1f1d450efd8cb3ec8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.293ex; height:2.509ex;" alt="{\displaystyle \xi \in E,}" loading="lazy"> welches keine Einheitswurzel in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist. Dann gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi ^{n}\neq 1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi ^{n}\neq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd8dd995bf9a64fe5fc63855b31da51abb9d2e76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.513ex; height:2.843ex;" alt="{\displaystyle \xi ^{n}\neq 1}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6b82f4024a32a795b10f00971e03ad88ebaa367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}" loading="lazy"> Dies Widerspricht der Endlichkeit der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a2566d01f104ef084ea424b8b35c2534f7f902b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.422ex; height:2.176ex;" alt="{\displaystyle E.}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Satz:_Verallgemeinerter_Dirichletscher_Einheitensatz">Satz: Verallgemeinerter Dirichletscher Einheitensatz</h4></div>

Sei die Situation wie zuvor. Dann gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc08a0b9b40bb460b541adc55bc55e08f3378f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle E(P)}" loading="lazy"> das <a href="Direktes_Produkt" title="Direktes Produkt">direkte Produkt</a> der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> aller <a href="Einheitswurzel" title="Einheitswurzel">Einheitswurzeln</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und einer Gruppe isomorph zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{s}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{s}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49f031b2c89a2db43ae8e44506b6b7e4e87e783a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.554ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{s}}" loading="lazy"> ist. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7903b8069a44c70f6f96511675bdd9a4ff200ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}" loading="lazy"> im Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\emptyset }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=\emptyset }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e931542551bba730105ec3f24f8f02149fad4760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.006ex; height:2.509ex;" alt="{\displaystyle P=\emptyset }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\operatorname {card} (P)-1=|P|-1,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=\operatorname {card} (P)-1=|P|-1,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46fd0d47e44859ca3906c458111353821b0ddc38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.933ex; height:2.843ex;" alt="{\displaystyle s=\operatorname {card} (P)-1=|P|-1,}" loading="lazy"> falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\neq \emptyset .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\neq \emptyset .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83c24d6343733cc3e46bfc1ec0a5619b3d9a30ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.653ex; height:2.843ex;" alt="{\displaystyle P\neq \emptyset .}" loading="lazy"> Ein Beweis dieser Aussage findet sich in <a href="#Literatur">Weil (1967)</a>, S. 78f. oder auch in <a href="#Literatur">Cassels (1967)</a>, S. 72f.

Satz: Dirichletscher Einheitensatz

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein algebraischer Zahlkörper und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/180e5aded86734744a16ec9c668976b5dbafd4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}" loading="lazy"> sein <a href="Ganzheitsring" title="Ganzheitsring">Ganzheitsring</a>. Dann gilt

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}^{\times }\cong \mu (K)\times \mathbb {Z} ^{r+s-1},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ae34ed5fe5677d4bde44c317b1dd4b1ff2c211d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.08ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}_{K}^{\times }\cong \mu (K)\times \mathbb {Z} ^{r+s-1},}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (K)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2abfb9cecc1c69b28f9ba8c6fd743a01626ee3fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.277ex; height:2.843ex;" alt="{\displaystyle \mu (K)}" loading="lazy"> die <a href="Endliche_Gruppe" title="Endliche Gruppe">endliche</a>, <a href="Zyklische_Gruppe" title="Zyklische Gruppe">zyklische Gruppe</a> der Einheitswurzeln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ist und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (r,s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f3e3778daed31bfa6149169529acb5d9fcc8b49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,s)}" loading="lazy"> der Typ des Zahlkörpers, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> ist die Anzahl der reellen Einbettungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac399360184e2035dbaea694e969a385cb16a6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.253ex; height:2.176ex;" alt="{\displaystyle 2s}" loading="lazy"> ist die Anzahl an komplexen Einbettungen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [K:\mathbb {Q} ]=r+2s.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [K:\mathbb {Q} ]=r+2s.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab4a6794fb048a2f632ee764850bb88ae743a79e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.992ex; height:2.843ex;" alt="{\displaystyle [K:\mathbb {Q} ]=r+2s.}" loading="lazy">

Bemerkung: Dies ist eine Verallgemeinerung des <a href="Dirichletscher_Einheitensatz" title="Dirichletscher Einheitensatz">Dirichletschen Einheitensatzes</a>. Für einen algebraischen Zahlkörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fea4b5006afa0eb81c21b577efe304ab545d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle K,}" loading="lazy"> setze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=P_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=P_{\infty }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b629774f471991e5e91672d882f376ae841c443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.212ex; height:2.509ex;" alt="{\displaystyle P=P_{\infty }}" loading="lazy"> um den Dirichletschen Einheitensatz in seiner klassischen Formulierung aus der verallgemeinerten Formulierung zu erhalten. In der englischen Literatur ist dieser Satz bekannt unter „Theorem of the units“. Natürlich ist der Dirichletsche Einheitensatz älter als obiges Resultat und wird im Allgemeinen zuvor eigenständig bewiesen und dann dazu benutzt, die Kompaktheit von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}^{1}/K^{\times }}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}^{1}/K^{\times }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61c17d9077d0c80c541fca2295306507bc2976ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:3.176ex;" alt="{\displaystyle \mathbb {A} _{K}^{1}/K^{\times }}" loading="lazy"> zu zeigen.

Beweis dieser Bemerkung:

Wir wissen bereits, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\mu (K).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\mu (K).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57b044a9e2fff3fd2be34eddc43fdb03213e7948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.798ex; height:2.843ex;" alt="{\displaystyle E=\mu (K).}" loading="lazy"> Weiterhin gilt, dass

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E(P_{\infty })&=K^{\times }\cap \left(\prod _{v\mid \infty }K_{v}^{\times }\times \prod _{v<\infty }{\mathcal {O}}_{v}^{\times }\right)\\&\cong K^{\times }\cap \prod _{v<\infty }{\mathcal {O}}_{v}^{\times }\\&\cong {\mathcal {O}}^{\times }.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a2be383676c0d9698af3cef2e50f570bad5ecb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:38.325ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}E(P_{\infty })&=K^{\times }\cap \left(\prod _{v\mid \infty }K_{v}^{\times }\times \prod _{v<\infty }{\mathcal {O}}_{v}^{\times }\right)\\&\cong K^{\times }\cap \prod _{v<\infty }{\mathcal {O}}_{v}^{\times }\\&\cong {\mathcal {O}}^{\times }.\end{aligned}}}" loading="lazy"></dd></dl>

Darüber hinaus gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |P_{\infty }|=r+s.}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |P_{\infty }|=r+s.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cac056e0e9ccc58f8bddbcaf4ac6da8cb530dd05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.386ex; height:2.843ex;" alt="{\displaystyle |P_{\infty }|=r+s.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="„Approximation_Theorems“">„Approximation Theorems“</h3></div>

Satz: „weak approximation theorem“

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{n},}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\cdot |_{n},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b1d5c5dcf5005cb0d15b264449a52a370f99a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.838ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{n},}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq n\leq N,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq n\leq N,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bc4315ef91e9cf483db1940c821ea5311f06aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.465ex; height:2.509ex;" alt="{\displaystyle 1\leq n\leq N,}" loading="lazy"> nichtäquivalente, nichttriviale Beträge auf einem Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}:=(K,|\cdot |_{n}).}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{n}:=(K,|\cdot |_{n}).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78f8d46686c567c3e1ebde68e0e181dedff19251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.684ex; height:3.009ex;" alt="{\displaystyle K_{n}:=(K,|\cdot |_{n}).}" loading="lazy"> Diese sind insbesondere topologische Räume. Bette <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> diagonal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{n=1}^{N}K_{n}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{n=1}^{N}K_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4133ead25f11c664ac0c784f5d5a666c77eb6f3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.091ex; height:3.509ex;" alt="{\displaystyle \textstyle \prod _{n=1}^{N}K_{n}}" loading="lazy"> ein. Dann gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> überall in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{n=1}^{N}K_{n}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{n=1}^{N}K_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4133ead25f11c664ac0c784f5d5a666c77eb6f3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.091ex; height:3.509ex;" alt="{\displaystyle \textstyle \prod _{n=1}^{N}K_{n}}" loading="lazy"> dicht ist. Mit anderen Worten gilt, dass für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/568095ad3924314374a5ab68fae17343661f2a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" loading="lazy"> und für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle (\alpha _{n})_{n}\in \prod _{n=1}^{N}K_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle (\alpha _{n})_{n}\in \prod _{n=1}^{N}K_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b4959be829f8b2af92e4f3169c47567d4be2dba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.666ex; height:3.509ex;" alt="{\displaystyle \textstyle (\alpha _{n})_{n}\in \prod _{n=1}^{N}K_{n}}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/453ca3ed19b7812c16718f3d75c2ae7bf796b9a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.937ex; height:2.509ex;" alt="{\displaystyle \xi \in K}" loading="lazy"> existiert, sodass

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\alpha _{n}-\xi |_{n}<\epsilon \qquad \forall n=1,\dotsc ,N.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb3e74b7764fb5e8d08713003f68b902274e9df9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.613ex; height:3.009ex;" alt="{\displaystyle |\alpha _{n}-\xi |_{n}<\epsilon \qquad \forall n=1,\dotsc ,N.}" loading="lazy"></dd></dl>

Ein Beweis dieser Aussage findet sich in <a href="#Literatur">Cassels (1967)</a>, S. 48f.

Eine Anwendung dieses Satzes befindet sich <a href="Tensorprodukt#Eine_Anwendung_in_der_algebraischen_Zahlentheorie" title="Tensorprodukt">hier</a>.

Satz: „strong approximation theorem“

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60faad24775635f4722ccc438093dbbfe05f34ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0}}" loading="lazy"> eine Stelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Definiere

<msup>

<mover>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V:={\widehat {\prod \limits _{v\neq v_{0}}}}^{{\mathcal {O}}_{v}}K_{v}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68f3e6a13b6267219eb62beeaa2054f1eb07967f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.239ex; height:7.176ex;" alt="{\displaystyle V:={\widehat {\prod \limits _{v\neq v_{0}}}}^{{\mathcal {O}}_{v}}K_{v}.}" loading="lazy"></dd></dl>

Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> dicht in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b2661a49b86bd1a5548e527bbfb068aa9f59978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}" loading="lazy"> Ein Beweis dieser Aussage findet sich in <a href="#Literatur">Cassels (1967)</a>, S. 67f.

Bemerkung: Der globale Körper ist diskret in seinem Adelering. Für dieses Resultat ist es wichtig, dass alle Stellen des globalen Körpers betrachtet werden. Der vorherige Satz zeigt, dass bereits das Weglassen von einer Stelle die Diskretheit in die Dichtheit des globalen Körpers verwandelt.

<div class="mw-heading mw-heading3"><h3 id="„Lokal-Global“_und_Hasse-Prinzip">„Lokal-Global“ und Hasse-Prinzip</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Lokal-Global-Prinzip" class="mw-disambig" title="Lokal-Global-Prinzip">Lokal-Global-Prinzip</a></div>

Begriffsmotivation: „Lokal“ und „Global“

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> eine endliche Körpererweiterung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle K.}" loading="lazy"> Dann bezeichnet man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> als globale Erweiterung. Sei nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> eine Stelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> eine über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> liegende Stelle von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d6358b7808f1224473a219f51e5eede2495fa88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle L.}" loading="lazy"> Dann bezeichnet man die (endliche) Körpererweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{w}/K_{v}}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{w}/K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c31a1748e08d130f4052ecc4e8e655e5dc15a124.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.157ex; height:2.843ex;" alt="{\displaystyle L_{w}/K_{v}}" loading="lazy"> als lokale Erweiterung. Woher kommen nun diese Bezeichnungen? Um dies einzusehen, betrachten wir den Funktionenkörperfall, bspw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {C} (t),}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {C} (t),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/035f45e4fd3d566031c381ba7cd2149370fcec4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.138ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {C} (t),}" loading="lazy"> obwohl dies kein globaler Körper mehr ist. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L/K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0381945929156997b99bb43e5b7067d18c9a84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}" loading="lazy"> eine endliche Körpererweiterung. Die Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> sind algebraische Funktionen auf einer <a href="Riemannsche_Fl%C3%A4che" title="Riemannsche Fläche">Riemannschen Fläche</a>, also auf einem globalen Objekt. Der Übergang von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{w}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{w}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f1f877323bcb0fab941ee2d2c30ddaa045ab844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.992ex; height:2.509ex;" alt="{\displaystyle L_{w}}" loading="lazy"> bedeutet nun, dass wir zu Potenzreihenentwicklungen übergehen, also zum lokalen Studium solcher Funktionen. Für mehr Informationen wird auf <a href="#Literatur">Neukirch (2007)</a>, S. 169 verwiesen.

Satz: Minkowski-Hasse

Eine <a href="Quadratische_Form" title="Quadratische Form">quadratische Form</a> über einem Zahlkörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> stellt genau dann die Null dar, wenn sie dies über jeder Komplettierung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> tut.

Bemerkung: Dies ist das Hasse-Prinzip für quadratische Formen, im Allgemeinen, d. h. für Polynome belieben Grades, ist das Hasse-Prinzip nicht gültig.

Bemerkung: Das Lokal-Global-Prinzip ist also jenes Prinzip, welches eine Problemstellung über einem Zahlkörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> auf analoge Problemstellungen über seinen Komplettierungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> zurückführt.

<div class="mw-heading mw-heading3"><h3 id="Charaktere_auf_dem_Adelering">Charaktere auf dem Adelering</h3></div>

Definition: Duale Gruppe

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine lokalkompakte, <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a>. Definiere die duale Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {G}}}">

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</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {G}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3257528da95b4ab702b74241ff5356d688de2ce0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:3.009ex;" alt="{\displaystyle {\widehat {G}}}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> als die Menge aller <a href="Charakter_(Mathematik)" title="Charakter (Mathematik)">Charaktere</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

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<annotation encoding="application/x-tex">{\displaystyle G,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> d. h. die Menge aller stetigen Gruppenhomomorphismen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/792104f18258be2ef324c99a99138e0ddbcb790c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.454ex; height:2.843ex;" alt="{\displaystyle \mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}.}" loading="lazy"> Auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {G}}}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {G}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3257528da95b4ab702b74241ff5356d688de2ce0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:3.009ex;" alt="{\displaystyle {\widehat {G}}}" loading="lazy"> wird die Topologie der <a href="Gleichm%C3%A4%C3%9Fige_Konvergenz" title="Gleichmäßige Konvergenz">gleichmäßigen Konvergenz</a> auf kompakten Teilmengen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> installiert. Man kann zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {G}}}">

<mover>

</mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {G}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3257528da95b4ab702b74241ff5356d688de2ce0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:3.009ex;" alt="{\displaystyle {\widehat {G}}}" loading="lazy"> wieder eine lokalkompakte, abelsche Gruppe wird.

Satz: Selbstdualität des Adelerings

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper. Der Adelering ist selbstdual, d. h. es gilt

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<mover>

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</mover>

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<mtext> stetiger </mtext>

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<mtext> Gruppenhomomorphismus </mtext>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}\cong {\widehat {\mathbb {A} _{K}}}:=\{\chi :\mathbb {A} _{K}\rightarrow \mathbb {T} ,\chi \,\,{\text{ ist }}{\text{ ein }}{\text{ stetiger }}{\text{ Gruppenhomomorphismus }}\}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9cb42be70467af5a59f4b4374e4b36a2e18e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.139ex; height:3.509ex;" alt="{\displaystyle \mathbb {A} _{K}\cong {\widehat {\mathbb {A} _{K}}}:=\{\chi :\mathbb {A} _{K}\rightarrow \mathbb {T} ,\chi \,\,{\text{ ist }}{\text{ ein }}{\text{ stetiger }}{\text{ Gruppenhomomorphismus }}\}.}" loading="lazy"></dd></dl>

Beweis: In einem ersten Schritt wird gezeigt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d4e4201ae0ff8ca59162a39dd4196c1a4077c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.509ex;" alt="{\displaystyle K_{v}}" loading="lazy"> selbstdual ist für jede Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,}">

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<annotation encoding="application/x-tex">{\displaystyle v,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7ba401aee0189f8031d21020a0c640a03339c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.774ex; height:2.009ex;" alt="{\displaystyle v,}" loading="lazy"> sofern man einen Charakter fixiert. Wir führen dies am Beispiel von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle K_{v}=\mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10dc704868f48c83bd0604bc3c757d5c615f4b5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.779ex; height:2.509ex;" alt="{\displaystyle K_{v}=\mathbb {R} }" loading="lazy"> vor. Definiere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\infty }:\mathbb {R} \rightarrow \mathbb {T} }">

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<annotation encoding="application/x-tex">{\displaystyle e_{\infty }:\mathbb {R} \rightarrow \mathbb {T} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7574bc2e095f0e843de97ffb362d6ea613b321f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.739ex; height:2.509ex;" alt="{\displaystyle e_{\infty }:\mathbb {R} \rightarrow \mathbb {T} }" loading="lazy"> via: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{\infty }(t):=\exp(2\pi it).}">

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<annotation encoding="application/x-tex">{\displaystyle e_{\infty }(t):=\exp(2\pi it).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79b97778bf607cf35916b120253caae86cefc47a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.498ex; height:2.843ex;" alt="{\displaystyle e_{\infty }(t):=\exp(2\pi it).}" loading="lazy"> Dann definiere die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :\mathbb {R} \rightarrow {\widehat {\mathbb {R} }},}">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {R} \rightarrow {\widehat {\mathbb {R} }},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1960f0023a815d61abe3ba63bfdcbb8e9cc87a2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.074ex; height:3.343ex;" alt="{\displaystyle \varphi :\mathbb {R} \rightarrow {\widehat {\mathbb {R} }},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\mapsto \varphi _{s},}">

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<annotation encoding="application/x-tex">{\displaystyle s\mapsto \varphi _{s},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2deba33fcc1bdd538a47a5a7a26407e32ad2832b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.875ex; height:2.343ex;" alt="{\displaystyle s\mapsto \varphi _{s},}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{s}:\mathbb {R} \rightarrow \mathbb {T} ,t\mapsto \exp(2\pi ist),}">

<msub>

</mrow>

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</mrow>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{s}:\mathbb {R} \rightarrow \mathbb {T} ,t\mapsto \exp(2\pi ist),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff4d702c19acd98945381c2a44f09f739b7f71de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.026ex; height:2.843ex;" alt="{\displaystyle \varphi _{s}:\mathbb {R} \rightarrow \mathbb {T} ,t\mapsto \exp(2\pi ist),}" loading="lazy"> also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{s}=e_{\infty }(\cdot s).}">

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</mrow>

</msub>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{s}=e_{\infty }(\cdot s).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04214c2facd37a5dec397b7564d2ec21d0f76f58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.774ex; height:2.843ex;" alt="{\displaystyle \varphi _{s}=e_{\infty }(\cdot s).}" loading="lazy"> Man zeigt schnell, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> ein Isomorphismus ist, welcher die Topologien respektiert. Hat man die Selbstdualität im lokalen gezeigt, kann man zeigen, dass der Adelering selbstdual ist, indem auf den lokalen Fall zurückgegriffen wird.

Satz: Algebraischer und topologischer Dual des Adelerings

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein globaler Körper und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">

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<annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656111758322ace96d80a9371771aa6d3de25437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }" loading="lazy"> ein nicht-trivialer Charakter von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K},}">

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</mrow>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bef6ca8bc0051c81dee17fa2d17e74c6029d22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.018ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K},}" loading="lazy"> welcher allerdings trivial auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> wirkt. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> ein endlich-dimensionaler <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">-Vektorraum. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{\star }}">

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</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle E^{\star }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12c6eebe1d81df59306aedd5918e4f4726360ca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.848ex; height:2.343ex;" alt="{\displaystyle E^{\star }}" loading="lazy"> der algebraische Dual von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,}">

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<annotation encoding="application/x-tex">{\displaystyle E,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89862747e88ca143e979241a9a243b5ef66ddc67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.422ex; height:2.509ex;" alt="{\displaystyle E,}" loading="lazy"> sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}^{\star }}">

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</mrow>

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</msubsup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}^{\star }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4d09bff21ffe60fab3c931f6b2e73efc80bf343.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.166ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{E}^{\star }}" loading="lazy"> der algebraische Dual von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03356310defcb93f5d83669bd76846c87f90ee3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}}" loading="lazy"> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}'}">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/529536c56fea4fb8d89a74d90fab4c04e46b18a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{E}'}" loading="lazy"> der topologische Dual von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}.}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ccdb1ba27215e0ae8d8fc61df305b7d4b90073f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.813ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{E}.}" loading="lazy"> Dann induziert die Abbildungsvorschrift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle e,e'\rangle =\chi ([e,e^{\star }])}">

<msup>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \langle e,e'\rangle =\chi ([e,e^{\star }])}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1620e9d3a77121cb3bc69d4785927eeeb8f59739.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.607ex; height:3.009ex;" alt="{\displaystyle \langle e,e'\rangle =\chi ([e,e^{\star }])}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in \mathbb {A} _{E}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\in \mathbb {A} _{E}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e29a1f31d1d7eefda054732f99555ba8ec2a0062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.09ex; height:2.509ex;" alt="{\displaystyle e\in \mathbb {A} _{E}}" loading="lazy"> einen Isomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\star }\mapsto e'}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\star }\mapsto e'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b402a3dacfe74e1617ec8b9e674eee8683abece1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.52ex; height:2.509ex;" alt="{\displaystyle e^{\star }\mapsto e'}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}^{\star }\rightarrow \mathbb {A} _{E}',}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}^{\star }\rightarrow \mathbb {A} _{E}',}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15d24573b8bb5fec9c8b6e3ddca53f8ee7ddcbab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.593ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{E}^{\star }\rightarrow \mathbb {A} _{E}',}" loading="lazy"> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e'\in \mathbb {A} _{E}'}">

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e'\in \mathbb {A} _{E}'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47ddd1fc2d7aa533390f9a0aa099742c87609964.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.775ex; height:3.009ex;" alt="{\displaystyle e'\in \mathbb {A} _{E}'}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }.}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7efdabd6c5a4035056575be72ae5962763ddc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.791ex; height:2.843ex;" alt="{\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }.}" loading="lazy"> Hierbei meint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\cdot },{\cdot }\rangle }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle {\cdot },{\cdot }\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/468a71a5490f2d634b4878381aadb2f9ccfc3f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle {\cdot },{\cdot }\rangle }" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\cdot },{\cdot }]}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [{\cdot },{\cdot }]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e58dc6e37720306f73f8f451fdf795d655f6a1f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [{\cdot },{\cdot }]}" loading="lazy"> jeweils die entsprechende bilineare Paarung auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}'}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d24e35ae5e6870198fe42a9e19df85a9be0ff4dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.172ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}'}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}^{\star }.}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}^{\star }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3f92da047ebd553ab2019d82e9278467a6ce3ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.819ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}^{\star }.}" loading="lazy"> Darüber hinaus gilt folgendes: Wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c1fe63ce1c982e8e1c3f85f707b197e6e728968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.144ex; height:2.843ex;" alt="{\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }}" loading="lazy"> zusätzlich noch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ([e,e^{\star }])=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi ([e,e^{\star }])=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2e1af4eb468d2e2427afdf023b1b0efca03c3cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.074ex; height:2.843ex;" alt="{\displaystyle \chi ([e,e^{\star }])=1}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34778736d9c6d607a4da3d25594b38dd3e8c82ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}" loading="lazy"> erfüllt, dann gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\star }\in E^{\star }.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{\star }\in E^{\star }.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8c257f19174dfa39ac48325de8f8bd8f1ed1c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.473ex; height:2.343ex;" alt="{\displaystyle e^{\star }\in E^{\star }.}" loading="lazy"> Ein Beweis dieser Aussage findet sich in <a href="#Literatur">Weil (1967)</a>, S. 66.

Mit Hilfe der Charaktere auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{K}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{K}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b88e22526256c10a2a2356a4f6bc9c7ad895bd86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle \mathbb {A} _{K}}" loading="lazy"> können wir <a href="Fourier-Analysis" title="Fourier-Analysis">Fourier-Analysis</a> auf dem Adelering betreiben (vgl. <a href="#Literatur">Deitmar (2010)</a>, S. 129ff).

<div class="mw-heading mw-heading3"><h3 id="Die_Doktorarbeit_von_John_Tate">Die Doktorarbeit von John Tate</h3></div>

<a href="John_T._Tate" title="John T. Tate">John Tate</a> erzielte in seiner Doktorarbeit „Fourier analysis in number fields and Heckes Zetafunctions“ (vgl. <a href="#Literatur">Cassels (1967)</a>) Erkenntnisse über <a href="L-Funktion" title="L-Funktion">L-Funktionen</a>, indem er Fourieranalyse auf den Adelering bzw. die Idelegruppe anwendete. Der Adelering und die Idelegruppe finden daher Anwendung bei der Untersuchung der <a href="Riemannsche_Zetafunktion" class="mw-redirect" title="Riemannsche Zetafunktion">Riemannschen Zetafunktion</a> und bei der Untersuchung allgemeiner Zetafunktionen bzw. L-Funktionen. Man kann diese Funktionen in einer adelischen Version definieren und sie als Integral über den Adelering bzw. die Idelegruppe über die entsprechenden Haarmaße darstellen und daraus meromorphe Fortsetzungen und Funktionalgleichungen ableiten. Wir geben ein Beispiel. Für jede komplexe Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13da7c2bac9dd6324ac83093fb42f3a9e86fbd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {C} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy"> gilt:

<msub>

<mover>

</mrow>

</mover>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{\widehat {\mathbb {Z} }}|x|^{s}d^{\times }x=\zeta (s),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20045a8dd30ffb0a7e83b3496f7f80069948168d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.434ex; height:5.676ex;" alt="{\displaystyle \int _{\widehat {\mathbb {Z} }}|x|^{s}d^{\times }x=\zeta (s),}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{\times }x}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d^{\times }x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c1a1cd1112c7f630305fc9edaad0b1030d86faf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.059ex; height:2.343ex;" alt="{\displaystyle d^{\times }x}" loading="lazy"> das eindeutig normalisierte Haarmaß auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}">

<msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f68751aa23c5257ad5f77a307adbc5feaacce4db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.104ex; height:2.843ex;" alt="{\displaystyle \mathbb {A} _{\mathbb {Q} ,fin}}" loading="lazy"> ist mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{\times }x({\widehat {\mathbb {Z} }}^{\times })=1,}">

<msup>

</mrow>

</msup>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d^{\times }x({\widehat {\mathbb {Z} }}^{\times })=1,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fde90d1b84e417f22659f1bfefe579b666162bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.837ex; height:3.509ex;" alt="{\displaystyle d^{\times }x({\widehat {\mathbb {Z} }}^{\times })=1,}" loading="lazy"> welches durch Null auf den ganzen Adelering ausgedehnt wird. Die obige Gleichung bedeutet, dass die Riemannsche Zetafunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> als Integral über den Adelering bzw. einer Teilmenge derselben dargestellt werden kann. Ein Beweis findet sich in <a href="#Literatur">Deitmar (2010)</a>, S. 128, Satz 5.3.4. Beachte außerdem S. 139ff für weitere Informationen über die Doktorarbeit von John Tate.

<div class="mw-heading mw-heading3"><h3 id="Automorphe_Formen">Automorphe Formen</h3></div>

Wir betrachten den Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} .}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b65039e866ecde3cb208fa49082994d3b17f938b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.619ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} .}" loading="lazy">

In moderner Notation ist eine <a href="Automorphe_Form" class="mw-redirect" title="Automorphe Form">automorphe Form</a> eine Funktion auf der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }),}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c47fb2b4c998de5511b9de11e71c410a04e961b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.976ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }),}" loading="lazy"> welche einige zusätzliche Bedingungen erfüllt. Um diese zu beschreiben, definieren wir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }):|\operatorname {det} (x)|=1\}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }):|\operatorname {det} (x)|=1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f08210f37403af376115335b4db72d7b9be479b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.011ex; height:3.176ex;" alt="{\displaystyle (\mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }):|\operatorname {det} (x)|=1\}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{\mathbb {R} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{\mathbb {R} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d6777518adc2421dbe596b7242d340e42d89a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.006ex; height:2.509ex;" alt="{\displaystyle Z_{\mathbb {R} }}" loading="lazy"> als das <a href="Zentrum_(Algebra)" title="Zentrum (Algebra)">Zentrum</a> der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} _{2}(\mathbb {R} ).}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} _{2}(\mathbb {R} ).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9277d605f1517e0f480707f32c3091fc08b13074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.466ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} _{2}(\mathbb {R} ).}" loading="lazy"> Es gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} _{2}(\mathbb {Q} )\backslash (\mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }))^{1}\cong (\mathrm {GL} _{2}(\mathbb {Q} )Z_{\mathbb {R} })\backslash \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }).}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} _{2}(\mathbb {Q} )\backslash (\mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }))^{1}\cong (\mathrm {GL} _{2}(\mathbb {Q} )Z_{\mathbb {R} })\backslash \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eae71d2deb3e04dcc2b25e9ca8b9b54726f3b522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.306ex; height:3.176ex;" alt="{\displaystyle \mathrm {GL} _{2}(\mathbb {Q} )\backslash (\mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }))^{1}\cong (\mathrm {GL} _{2}(\mathbb {Q} )Z_{\mathbb {R} })\backslash \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} }).}" loading="lazy"> Wir definieren eine automorphe Form als ein Element des Vektorraums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}((\mathrm {GL} _{2}(\mathbb {Q} )Z_{\mathbb {R} })\backslash \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })).}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}((\mathrm {GL} _{2}(\mathbb {Q} )Z_{\mathbb {R} })\backslash \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad0972dd2543362a7d5349361871906979007180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.35ex; height:3.176ex;" alt="{\displaystyle L^{2}((\mathrm {GL} _{2}(\mathbb {Q} )Z_{\mathbb {R} })\backslash \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })).}" loading="lazy"> Um automorphe Formen zu untersuchen, ist es wichtig die Darstellungen der Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6909a63ad9c935fe479da04d3590f45a3059c1f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.33ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}" loading="lazy"> zu kennen, welche durch den Tensorproduktsatz charakterisiert werden. Man kann außerdem auch sogenannte automorphe L-Funktionen betrachten, welche als Integral über die Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6909a63ad9c935fe479da04d3590f45a3059c1f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.33ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}" loading="lazy"> dargestellt werden können. Weitere Informationen finden sich in <a href="#Literatur">Deitmar (2010)</a> in dem Kapitel über die automorphen Darstellungen der Adelegruppe und in dem Kapitel über die automorphen L-Funktionen.

<div class="mw-heading mw-heading3"><h3 id="Weitere_Anwendungen">Weitere Anwendungen</h3></div>

Verallgemeinerungen des <a href="Artinsches_Reziprozit%C3%A4tsgesetz" title="Artinsches Reziprozitätsgesetz">Artinschen Reziprozitätsgesetzes</a> führen zur Verbindung von automorphen Darstellungen und <a href="Galois-Darstellung" title="Galois-Darstellung">Galois-Darstellungen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> (<a href="Langlands-Programm" title="Langlands-Programm">Langlands-Programm</a>).

Die <a href="Idelegruppe" title="Idelegruppe">Idelegruppe</a>, speziell die <a href="Idelklassengruppe" class="mw-redirect" title="Idelklassengruppe">Idelklassengruppe</a>, findet Anwendung in der <a href="Klassenk%C3%B6rpertheorie" title="Klassenkörpertheorie">Klassenkörpertheorie</a>, welche sich mit <a href="Abelsche_K%C3%B6rpererweiterung" class="mw-redirect" title="Abelsche Körpererweiterung">abelschen Körpererweiterungen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> beschäftigt. Das Produkt der lokalen Reziprozitätskarten in der Klassenkörpertheorie gibt einen <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismus</a> von der Idelegruppe in die <a href="Galoisgruppe" title="Galoisgruppe">Galoisgruppe</a> der maximalen abelschen Erweiterung über einem <a href="Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper">algebraischen Zahlkörper</a>. Das Artinsche Reziprozitätsgesetz, welches eine Verallgemeinerung des <a href="Quadratisches_Reziprozit%C3%A4tsgesetz" title="Quadratisches Reziprozitätsgesetz">quadratischen Reziprozitätsgesetzes</a> von Gauß ist, besagt, dass das Produkt in der multiplikativen Gruppe des Zahlkörpers verschwindet. Daher erhalten wir die globale Reziprozitätskarte der Idelklassengruppe von dem abelschen Teil der absoluten Galoisgruppe der Körpererweiterung.

Auch außerhalb der Klassenkörpertheorie finden sich Anwendungen. Die Selbstdualität des Adelerings impliziert im Funktionenkörperfall (hier ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> ein Funktionenkörper über einer <a href="Algebraische_Kurve" title="Algebraische Kurve">Kurve</a>) den <a href="Satz_von_Riemann-Roch" title="Satz von Riemann-Roch">Satz von Riemann-Roch</a> für diese Kurve und die entsprechende duale Theorie für diese Kurve.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="John_Cassels" title="John Cassels">John Cassels</a>, <a href="Albrecht_Fr%C3%B6hlich" title="Albrecht Fröhlich">Albrecht Fröhlich</a>: Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute). Academic Press, London 1987, ISBN 0-12-163251-2.</li>

<li><a href="J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Jürgen Neukirch</a>: Algebraische Zahlentheorie. unveränd. Nachdruck der 1. Auflage. Springer, Berlin 2007, ISBN 978-3-540-37547-0.</li>

<li><a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a>: Basic number theory. Springer, Berlin / Heidelberg / New York 1967, ISBN 978-3-662-00048-9.</li>

<li>Anton Deitmar: Automorphe Formen. Springer, Berlin / Heidelberg u. a. 2010, ISBN 978-3-642-12389-4.</li>

<li><a href="Serge_Lang" title="Serge Lang">Serge Lang</a>: Algebraic number theory, Graduate Texts in Mathematics 110. 2. Auflage. Springer-Verlag, New York 1994, ISBN 0-387-94225-4.</li></ul></div><div><div class="zim-footer">

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