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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Divisionsalgebra</span></h1>
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<td><b>Divisionsalgebra</b>
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<p>berührt die Spezialgebiete
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<ul><li><a href="Mathematik" title="Mathematik">Mathematik</a>
<ul><li><a href="Abstrakte_Algebra" title="Abstrakte Algebra">Abstrakte Algebra</a></li>
<li><a href="Lineare_Algebra" title="Lineare Algebra">Lineare Algebra</a></li></ul></li></ul>
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<p>ist Spezialfall von
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<ul><li><a href="Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebra</a></li>
<li><a href="Tern%C3%A4rk%C3%B6rper" title="Ternärkörper">Quasikörper</a> (für Divisionsalgebra mit Eins)</li></ul>
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<p>umfasst als Spezialfälle
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<ul><li><a href="Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörper</a></li>
<li><a href="Oktave_(Mathematik)" title="Oktave (Mathematik)">Oktaven</a></li></ul>
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<p><b>Divisionsalgebra</b> ist ein Begriff aus dem <a href="Mathematik" title="Mathematik">mathematischen</a> Teilgebiet der <a href="Abstrakte_Algebra" title="Abstrakte Algebra">abstrakten Algebra</a>. Grob gesprochen handelt es sich bei einer Divisionsalgebra um einen <a href="Vektorraum" title="Vektorraum">Vektorraum</a>, in dem man Elemente multiplizieren und dividieren kann.
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<div class="mw-heading mw-heading2"><h2 id="Definition_und_Beispiel">Definition und Beispiel</h2></div>
<p>Eine Divisionsalgebra ist eine nicht notwendigerweise <a href="Assoziative_Algebra" title="Assoziative Algebra">assoziative</a> <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">Algebra</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 D\neq \{0\}}">
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<annotation encoding="application/x-tex">{\displaystyle a\times x=b}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy"></span> die Vektormultiplikation in der Algebra. Das ist im endlich-dimensionalen Fall gleichbedeutend damit, dass die Algebra frei von <a href="Nullteiler" title="Nullteiler">Nullteilern</a> ist.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Enthält die Divisionsalgebra ein Einselement, so dass 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 D}">
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<annotation encoding="application/x-tex">{\displaystyle a\in D}</annotation>
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<annotation encoding="application/x-tex">{\displaystyle a\times 1=1\times a=a}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e53bdc459fa735e3a18f2b06d00ba2a479fa06b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.892ex; height:2.176ex;" alt="{\displaystyle a\times 1=1\times a=a}" loading="lazy"></span>, so spricht man von einer Divisionsalgebra mit Eins.
</p><p>Beispiel einer Divisionsalgebra ohne Einselement mit den beiden Einheiten <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 e_{1}}">
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</p><p><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 {\begin{matrix}e_{1}\times e_{1}&=&+e_{1}\\e_{1}\times e_{2}&=&-e_{2}\\e_{2}\times e_{1}&=&-e_{2}\\e_{2}\times e_{2}&=&-e_{1}\end{matrix}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}e_{1}\times e_{1}&=&+e_{1}\\e_{1}\times e_{2}&=&-e_{2}\\e_{2}\times e_{1}&=&-e_{2}\\e_{2}\times e_{2}&=&-e_{1}\end{matrix}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d062ec7d4c671602521ce81867de30a68afacaca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:18.266ex; height:12.509ex;" alt="{\displaystyle {\begin{matrix}e_{1}\times e_{1}&=&+e_{1}\\e_{1}\times e_{2}&=&-e_{2}\\e_{2}\times e_{1}&=&-e_{2}\\e_{2}\times e_{2}&=&-e_{1}\end{matrix}}}" loading="lazy"></span>
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<div class="mw-heading mw-heading2"><h2 id="Sätze_über_reelle_Divisionsalgebren"><span id="S.C3.A4tze_.C3.BCber_reelle_Divisionsalgebren"></span>Sätze über reelle Divisionsalgebren</h2></div>
<p>Eine endlichdimensionale Divisionsalgebra über den <a href="Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reellen Zahlen</a> hat stets die <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> 1, 2, 4 oder 8. Das wurde 1958 mit <a href="Algebraische_Topologie" title="Algebraische Topologie">topologischen Methoden</a> von <a href="John_Milnor" class="mw-redirect" title="John Milnor">John Milnor</a> und <a href="Michel_Kervaire" title="Michel Kervaire">Michel Kervaire</a> bewiesen.
</p><p>Die vier reellen, <a href="Normierte_Algebra" title="Normierte Algebra">normierten</a>, Divisionsalgebren mit Eins sind (bis auf <a href="Isomorphismus" title="Isomorphismus">Isomorphie</a>):
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<ul><li>die reellen Zahlen selbst</li>
<li>die <a href="Komplexe_Zahlen" class="mw-redirect" title="Komplexe Zahlen">komplexen Zahlen</a></li>
<li>die <a href="Quaternion" title="Quaternion">Quaternionen</a></li>
<li>die <a href="Oktave_(Mathematik)" title="Oktave (Mathematik)">Oktaven</a> auch Oktonionen oder Cayley-Zahlen.</li></ul>
<p>Dieses Resultat ist als Satz von <a href="Adolf_Hurwitz" title="Adolf Hurwitz">Hurwitz</a> (1898) bekannt. Alle außer den Oktaven erfüllen das <a href="Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetz</a> der Multiplikation.
</p><p>Jede reelle, endlichdimensionale und assoziative Divisionsalgebra ist isomorph zu den reellen Zahlen, den komplexen Zahlen oder zu den Quaternionen; dies ist der <a href="Satz_von_Frobenius_(reelle_Divisionsalgebren)" title="Satz von Frobenius (reelle Divisionsalgebren)">Satz von Frobenius</a> (1877).
</p><p>Jede reelle, endlichdimensionale kommutative Divisionsalgebra hat maximal die Dimension 2 als Vektorraum über den reellen Zahlen (Satz von Hopf, <a href="Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a> 1940). Dabei wird Assoziativität nicht vorausgesetzt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Topologische_Beweise_der_Existenz_von_Divisionsalgebren_über_den_reellen_Zahlen"><span id="Topologische_Beweise_der_Existenz_von_Divisionsalgebren_.C3.BCber_den_reellen_Zahlen"></span>Topologische Beweise der Existenz von Divisionsalgebren über den reellen Zahlen</h2></div>
<p>Heinz Hopf zeigte 1940, dass die Dimension einer Divisionsalgebra eine Potenz von 2 sein muss.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 1958 zeigten dann <a href="Michel_Kervaire" title="Michel Kervaire">Michel Kervaire</a> und <a href="John_Milnor" class="mw-redirect" title="John Milnor">John Milnor</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> unabhängig voneinander unter Benutzung des Periodizitätssatzes von <a href="Raoul_Bott" title="Raoul Bott">Raoul Bott</a> über Homotopiegruppen der <a href="Unit%C3%A4re_Gruppe" title="Unitäre Gruppe">unitären</a> und <a href="Orthogonale_Gruppe" title="Orthogonale Gruppe">orthogonalen</a> Gruppen, dass die Dimensionen <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 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><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"></span>, <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 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/901fc910c19990d0dbaaefe4726ceb1a4e217a0f.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 2}" loading="lazy"></span>, <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 4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>4</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 4}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/295b4bf1de7cd3500e740e0f4f0635db22d87b42.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 4}" loading="lazy"></span> oder <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 8}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>8</mn>
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<annotation encoding="application/x-tex">{\displaystyle 8}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5.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 8}" loading="lazy"></span> sein müssen (entsprechend den reellen Zahlen, den komplexen Zahlen, den Quaternionen und Oktonionen). Letztere Aussage konnte bisher nicht rein algebraisch bewiesen werden. Der Beweis wurde von <a href="Michael_Atiyah" class="mw-redirect" title="Michael Atiyah">Michael Atiyah</a> und <a href="Friedrich_Hirzebruch" title="Friedrich Hirzebruch">Friedrich Hirzebruch</a> auch mit Hilfe der <a href="Topologische_K-Theorie" title="Topologische K-Theorie">K-Theorie</a> formuliert.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p><p>Dazu betrachtet man nach Hopf die Multiplikation einer Divisionsalgebra der Dimension <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span> über den reellen Zahlen als stetige Abbildung <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 {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/803bca6950fe01d404d3f4fd04af0a87e53fdee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.144ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}" loading="lazy"></span> oder eingeschränkt auf Elemente der Länge <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 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>
</semantics>
</math></span><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"></span> (man teile durch die Norm der Elemente, diese ist ungleich null für Elemente ungleich null da eine Divisionsalgebra nullteilerfrei ist) als Abbildung <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 S^{n-1}\times S^{n-1}\to S^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}\times S^{n-1}\to S^{n-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36eb651485aebed0f11b4bd55791f34ef569c71a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.976ex; height:2.676ex;" alt="{\displaystyle S^{n-1}\times S^{n-1}\to S^{n-1}}" loading="lazy"></span>. Hopf bewies, dass es eine solche ungerade Abbildung (das heißt <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 f(-x,y)=-f(x,y)=f(x,-y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(-x,y)=-f(x,y)=f(x,-y)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4df44fa75540979e501d9c11db6df4f914b5f66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.442ex; height:2.843ex;" alt="{\displaystyle f(-x,y)=-f(x,y)=f(x,-y)}" loading="lazy"></span>) nur gibt, wenn <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span> eine Potenz 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 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/901fc910c19990d0dbaaefe4726ceb1a4e217a0f.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 2}" loading="lazy"></span> ist. Dazu benutzte er die Homologiegruppen des projektiven Raums. Es gibt weitere äquivalente Formulierungen zur Existenz von Divisionsalgebren der Dimension <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span>:
</p>
<ul><li>Die Sphäre <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 S^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/742bebb03fe630674b18823a59d2c75efd0066e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}" loading="lazy"></span> (oder der projektive Raum <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 {P} ^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90c2f180f9a2da4a5d07b37109de8804da38385d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.739ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{n-1}}" loading="lazy"></span>) ist parallelisierbar (das heißt, es gibt zu jedem Punkt <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 x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> 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 S^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/742bebb03fe630674b18823a59d2c75efd0066e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}" loading="lazy"></span> (n-1) <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängige</a> Vektoren, die stetig 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 x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> abhängen und senkrecht auf <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 x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> stehen).</li>
<li>Es gibt Vektorraumbündel <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 E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><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"></span> über <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 S^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/742bebb03fe630674b18823a59d2c75efd0066e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}" loading="lazy"></span> mit <a href="Stiefel-Whitney-Klassen" title="Stiefel-Whitney-Klassen">Stiefel-Whitney Kohomologieklasse</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 w_{n}(E)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle w_{n}(E)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52b8cafcaa2bcbce4cfcb1f03731058e6fe8e229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.468ex; height:2.843ex;" alt="{\displaystyle w_{n}(E)}" loading="lazy"></span> ungleich null.</li>
<li>Es gibt eine Abbildung <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 f\colon S^{2n-1}\to S^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon S^{2n-1}\to S^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4545d37a4bccf60105da86cbcbef8e8eba4b771.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.329ex; height:3.009ex;" alt="{\displaystyle f\colon S^{2n-1}\to S^{n}}" loading="lazy"></span> mit ungerader <a href="Hopf-Invariante" class="mw-redirect" title="Hopf-Invariante">Hopf-Invariante</a> (siehe <a href="Hopf-Verschlingung" title="Hopf-Verschlingung">Hopf-Verschlingung</a>). <a href="John_Frank_Adams" title="John Frank Adams">Frank Adams</a> zeigte, dass solche Abbildungen nur für <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 n\in \{2,4,8\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>2</mn>
<mo>,</mo>
<mn>4</mn>
<mo>,</mo>
<mn>8</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\in \{2,4,8\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02df37b60dca5b0bb10a29324f03b2361ccfdb34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.116ex; height:2.843ex;" alt="{\displaystyle n\in \{2,4,8\}}" loading="lazy"></span> existieren.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Anwendung">Anwendung</h2></div>
<ul><li>Divisionsalgebren mit Einselement sind <a href="Tern%C3%A4rk%C3%B6rper" title="Ternärkörper">Quasikörper</a> (nicht unbedingt umgekehrt). Daher liefert jedes Beispiel einer Divisionsalgebra <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 D}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> in der <a href="Synthetische_Geometrie" title="Synthetische Geometrie">synthetischen Geometrie</a> ein Beispiel für eine <a href="Affine_Translationsebene" title="Affine Translationsebene">Affine Translationsebene</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 D^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a1e7d554748b105413b259fdfb8c9587acb9f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Hyperkomplexe_Zahl" title="Hyperkomplexe Zahl">Hyperkomplexe Zahl</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Ebbinghaus et al.: <i>Zahlen</i>. Berlin: Springer, 1992, ISBN 3-540-55654-0</li>
<li>Stefaan Caenepeel, A. Verschoren <i>Rings, Hopf Algebras, and Brauer Groups</i>, CRC Press, 1998, ISBN 0-82470-153-4</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">z. B. Shafarevich, Grundzüge der algebraischen Geometrie, Vieweg 1972, S. 201. Die lineare Abbildung <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 \phi (x)=a\times x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>a</mi>
<mo>×<!-- × --></mo>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (x)=a\times x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edb944a05302c7ddf9e299eca522e0bd8a0133de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.023ex; height:2.843ex;" alt="{\displaystyle \phi (x)=a\times x}" loading="lazy"></span> (analog für Rechtsmultiplikation) bildet D auf sich ab und ist injektiv, der <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> besteht danach nur aus der Null.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Hopf, Ein topologischer Beitrag zur reellen Algebra, Comm. Math. Helvetici, Band 13, 1940/41, S. 223–226</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Milnor, Some consequences of a theorem of Bott, Annals of Mathematics, Band 68, 1958, S. 444–449</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Atiyah, Hirzebruch, Bott periodicity and the parallelisability of the spheres, Proc. Cambridge Phil. Soc., Band 57, 1961, S. 223–226</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Die Darstellung zu den topologischen Beweisen folgt Friedrich Hirzebruch, Divisionsalgebren und Topologie (Kapitel 10), in Ebbinghaus u. a. Zahlen, Springer, 1983</span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Adams, On the non-existence of elements of Hopf invariant one, Annals of Mathematics, Band 72, 1960, S. 20–104</span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Ein Beweis mit K-Theorie ist in Atiyah, K-Theory, Benjamin 1967</span>
</li>
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