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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Globaler Körper</span></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"><p><b>Globale Körper</b> sind die zentralen Studienobjekte des <a href="Mathematik" title="Mathematik">mathematischen</a> Teilgebietes der <a href="Algebraische_Zahlentheorie" title="Algebraische Zahlentheorie">algebraischen Zahlentheorie</a>. Der bekannteste globale <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> ist der der <a href="Rationale_Zahl" title="Rationale Zahl">rationalen Zahlen</a>. Demgegenüber entstehen <a href="Lokaler_K%C3%B6rper" title="Lokaler Körper">lokale Körper</a> durch Vervollständigungen globaler Körper.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Als globale Körper bezeichnet man
</p>
<ul><li>einerseits <a href="Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper">algebraische Zahlkörper</a>, d. h. endliche <a href="K%C3%B6rpererweiterung" title="Körpererweiterung">Erweiterungen</a> des <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körpers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>
</semantics>
</math></span><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"></span> der <a href="Rationale_Zahl" title="Rationale Zahl">rationalen Zahlen</a>. Sie haben Charakteristik Null.</li>
<li>und andererseits algebraische <a href="Funktionenk%C3%B6rper" title="Funktionenkörper">Funktionenkörper</a> positiver <a href="Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> vom <a href="Transzendenzgrad" class="mw-redirect" title="Transzendenzgrad">Transzendenzgrad</a> 1, d. h. endliche Erweiterungen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}(T)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}(T)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91777ddcd1e21455f7ab77cca73afc529466b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.925ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{p}(T)}" loading="lazy"></span> für eine <a href="Primzahl" title="Primzahl">Primzahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><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"></span> und eine <a href="Unbestimmte" title="Unbestimmte">Unbestimmte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span>.</li></ul>
<p>Die <a href="Vollst%C3%A4ndiger_Raum" title="Vollständiger Raum">Vervollständigungen</a> globaler Körper an jeder Stelle bezüglich ihrer jeweiligen Metriken sind lokale Körper. Dass sowohl Zahlkörper als auch Funktionenkörper globale Körper sind, drückt eine schon seit dem 19. Jahrhundert (<a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> u. a.) bekannte Analogie zwischen Zahl- und Funktionenkörpern aus. Diese ermöglicht es für den schwierigeren Zahlkörperfall häufig mit Methoden zu arbeiten, die im Funktionenkörperfall entwickelt wurden und dort eine natürliche geometrische Interpretation haben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Axiomatische_Charakterisierung_nach_Artin_und_Whaples">Axiomatische Charakterisierung nach <a href="Emil_Artin" title="Emil Artin">Artin</a> und <a href="George_Whaples" title="George Whaples">Whaples</a></h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><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"></span> ein Körper mit einer Menge von Primstellen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {V}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">V</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {V}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/548a9a5c6d88d76f8abb63d975edb43d4cf98e7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {V}}}" loading="lazy"></span>, sodass folgende Axiome erfüllt sind.
</p>
<ul><li>Für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K^{\times }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>×<!-- × --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\in K^{\times }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d8ccf296f9512533a139ae2dcca59b4508ce832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.675ex; height:2.343ex;" alt="{\displaystyle a\in K^{\times }}" loading="lazy"></span> ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|_{v}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>a</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |a|_{v}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1c43a165d4e1dca5625d02d5be54d4b55a9ec5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.814ex; height:3.009ex;" alt="{\displaystyle |a|_{v}=1}" loading="lazy"></span> für <a href="Fast_alle" title="Fast alle">fast alle</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in {\mathfrak {V}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">V</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v\in {\mathfrak {V}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a317a3ad5f2a6b84e46f1a27f856968417603b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.9ex; height:2.176ex;" alt="{\displaystyle v\in {\mathfrak {V}}}" loading="lazy"></span> und es gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{v\in {\mathfrak {V}}}|a|_{v}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">V</mi>
</mrow>
</mrow>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>a</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \prod _{v\in {\mathfrak {V}}}|a|_{v}=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7dc78d9bed9c38fbfc239279f7e9affeea5632ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.46ex; height:5.676ex;" alt="{\displaystyle \prod _{v\in {\mathfrak {V}}}|a|_{v}=1}" loading="lazy"></span> (Produktformel).</li>
<li>Es gibt ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in {\mathfrak {V}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">V</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v\in {\mathfrak {V}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a317a3ad5f2a6b84e46f1a27f856968417603b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.9ex; height:2.176ex;" alt="{\displaystyle v\in {\mathfrak {V}}}" loading="lazy"></span>, sodass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>v</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{v}}</annotation>
</semantics>
</math></span><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"></span> ein <a href="Lokaler_K%C3%B6rper" title="Lokaler Körper">lokaler Körper</a> ist.</li></ul>
<p>Dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><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"></span> ein globaler Körper und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {V}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">V</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {V}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/548a9a5c6d88d76f8abb63d975edb43d4cf98e7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {V}}}" loading="lazy"></span> besteht aus allen Primstellen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><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"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://www.williamstein.org/papers/ant/html/node81.html">William Stein, Global Fields</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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