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<div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:%C3%9Cbersicht_K%C3%B6rper.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/%C3%9Cbersicht_K%C3%B6rper.svg/400px-%C3%9Cbersicht_K%C3%B6rper.svg.png" decoding="async" width="400" height="309" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/%C3%9Cbersicht_K%C3%B6rper.svg/600px-%C3%9Cbersicht_K%C3%B6rper.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/%C3%9Cbersicht_K%C3%B6rper.svg/800px-%C3%9Cbersicht_K%C3%B6rper.svg.png 2x" data-file-width="848" data-file-height="656" /></a><figcaption>Körper im Zusammenhang mit ausgewählten mathematischen Teilgebieten (<a href="/wiki/Klassendiagramm" title="Klassendiagramm">Klassendiagramm</a>)</figcaption></figure> Ein Körper ist im <a href="/wiki/Teilgebiete_der_Mathematik" title="Teilgebiete der Mathematik">mathematischen Teilgebiet</a> der <a href="/wiki/Algebra" title="Algebra">Algebra</a> eine ausgezeichnete <a href="/wiki/Algebraische_Struktur" title="Algebraische Struktur">algebraische Struktur</a>, in der die <a href="/wiki/Addition" title="Addition">Addition</a>, <a href="/wiki/Subtraktion" title="Subtraktion">Subtraktion</a>, <a href="/wiki/Multiplikation" title="Multiplikation">Multiplikation</a> und <a href="/wiki/Division_(Mathematik)" title="Division (Mathematik)">Division</a> auf eine bestimmte Weise durchgeführt werden können. Die Bezeichnung „Körper“ wurde im 19. Jahrhundert von <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> eingeführt. Die wichtigsten Körper, die in fast allen Gebieten der Mathematik benutzt werden, sind der Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" 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} }"> der <a href="/wiki/Rationale_Zahl" title="Rationale Zahl">rationalen Zahlen</a>, der Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }"> der <a href="/wiki/Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a> und der Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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} }"> der <a href="/wiki/Komplexe_Zahl" title="Komplexe Zahl">komplexen Zahlen</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Formale_Definition">1 Formale Definition</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Allgemeine_Definition">1.1 Allgemeine Definition</a></li> <li class="toclevel-2 tocsection-3"><a href="#Einzelaufzählung_der_benötigten_Axiome">1.2 Einzelaufzählung der benötigten Axiome</a></li> <li class="toclevel-2 tocsection-4"><a href="#Definition_als_spezieller_Ring">1.3 Definition als spezieller Ring</a></li> <li class="toclevel-2 tocsection-5"><a href="#Bemerkungen">1.4 Bemerkungen</a></li> <li class="toclevel-2 tocsection-6"><a href="#Verallgemeinerungen:_Schiefkörper_und_Koordinatenkörper">1.5 Verallgemeinerungen: Schiefkörper und Koordinatenkörper</a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Eigenschaften_und_Begriffe">2 Eigenschaften und Begriffe</a></li> <li class="toclevel-1 tocsection-8"><a href="#Körpererweiterung">3 Körpererweiterung</a></li> <li class="toclevel-1 tocsection-9"><a href="#Beispiele">4 Beispiele</a></li> <li class="toclevel-1 tocsection-10"><a href="#Endliche_Körper">5 Endliche Körper</a></li> <li class="toclevel-1 tocsection-11"><a href="#Geschichte">6 Geschichte</a></li> <li class="toclevel-1 tocsection-12"><a href="#Siehe_auch">7 Siehe auch</a></li> <li class="toclevel-1 tocsection-13"><a href="#Literatur">8 Literatur</a></li> <li class="toclevel-1 tocsection-14"><a href="#Weblinks">9 Weblinks</a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise">10 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Formale_Definition">Formale Definition</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=1" title="Abschnitt bearbeiten: Formale Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Formale Definition">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Allgemeine_Definition">Allgemeine Definition</h3>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=2" title="Abschnitt bearbeiten: Allgemeine Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Allgemeine Definition">Quelltext bearbeiten</a>]</div> Ein Körper ist eine Menge <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}">, versehen mit zwei <a href="/wiki/Innere_zweistellige_Verkn%C3%BCpfung" class="mw-redirect" title="Innere zweistellige Verknüpfung">inneren zweistelligen Verknüpfungen</a> „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}">“ und „<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }">“ (die Addition und Multiplikation genannt werden), für die die folgenden Bedingungen, die Körperaxiome, erfüllt sind: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(K,+\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>K</mi> <mo>,</mo> <mo>+</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(K,+\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2201a2bcf1baad7cf8bd23315a6c464cfadfc160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.717ex; height:2.843ex;" alt="{\displaystyle \left(K,+\right)}"> ist eine <a href="/wiki/Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a> mit dem neutralen Element „0“.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}K\setminus \{0\},\cdot {\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}K\setminus \{0\},\cdot {\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4eb06f4313cac1dadf0f0c5a03e068ab8ddf18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.559ex; height:3.176ex;" alt="{\displaystyle {\bigl (}K\setminus \{0\},\cdot {\bigr )}}"> ist eine <a href="/wiki/Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppe</a> mit dem neutralen Element „1“.</li> <li>Ferner gilt das <a href="/wiki/Distributivgesetz" title="Distributivgesetz">Distributivgesetz</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot \left(b+c\right)=a\cdot b+a\cdot c\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot \left(b+c\right)=a\cdot b+a\cdot c\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edf34e0e04e8710d13188b7064881cb7f5aee4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.711ex; height:2.843ex;" alt="{\displaystyle a\cdot \left(b+c\right)=a\cdot b+a\cdot c\,}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a+b\right)\cdot c=a\cdot c+b\cdot c\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a+b\right)\cdot c=a\cdot c+b\cdot c\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c549a9cca2718e718d775c3c27b4bbd1992f382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.488ex; height:2.843ex;" alt="{\displaystyle \left(a+b\right)\cdot c=a\cdot c+b\cdot c\,}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e030a1e5672b28c3b53d974ce2c4e72b03589ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.209ex; height:2.509ex;" alt="{\displaystyle a,b,c\in K}">.</dd></dl></li></ol> <div class="mw-heading mw-heading3"><h3 id="Einzelaufzählung_der_benötigten_Axiome">Einzelaufzählung der benötigten Axiome</h3>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=3" title="Abschnitt bearbeiten: Einzelaufzählung der benötigten Axiome" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Einzelaufzählung der benötigten Axiome">Quelltext bearbeiten</a>]</div> Ein Körper muss also folgende Einzelaxiome erfüllen: <ol><li>Additive Eigenschaften: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b+c)=(a+b)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b+c)=(a+b)+c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44038eb287a7d11c82ecf1642362bff63a012b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle a+(b+c)=(a+b)+c}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e030a1e5672b28c3b53d974ce2c4e72b03589ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.209ex; height:2.509ex;" alt="{\displaystyle a,b,c\in K}"> (<a href="/wiki/Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetz</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ff3a49d65fc590e33a74fd613900dd5924d6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.168ex; height:2.509ex;" alt="{\displaystyle a,b\in K}"> (<a href="/wiki/Kommutativgesetz" title="Kommutativgesetz">Kommutativgesetz</a>)</li> <li>Es gibt ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f6d135a64287ecfe6ac6cd2219e57585c726c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.069ex; height:2.176ex;" alt="{\displaystyle 0\in K}">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+a=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba565116cf2f2d984f7b8365b054b70eb8f89308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle 0+a=a}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d838bfcfb39f7a33ffe31cd1c2a989b8ca3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle a\in K}"> (<a href="/wiki/Neutrales_Element" title="Neutrales Element">neutrales Element</a>).</li> <li>Zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d838bfcfb39f7a33ffe31cd1c2a989b8ca3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle a\in K}"> existiert ein additives <a href="/wiki/Inverses_Element" title="Inverses Element">Inverses</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)+a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)+a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/843fca7f0f8dbe1ca4a56fe5d11b705926ed4977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.178ex; height:2.843ex;" alt="{\displaystyle (-a)+a=0}">.</li></ol></li> <li>Multiplikative Eigenschaften: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be34a5bcecdbbd7f3d5a983e34f00bf0b80c5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.902ex; height:2.843ex;" alt="{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e030a1e5672b28c3b53d974ce2c4e72b03589ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.209ex; height:2.509ex;" alt="{\displaystyle a,b,c\in K}"> (<a href="/wiki/Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetz</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=b\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=b\cdot a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b7dede7493e0231b3ad6ff9b54f4eae954108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.911ex; height:2.176ex;" alt="{\displaystyle a\cdot b=b\cdot a}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ff3a49d65fc590e33a74fd613900dd5924d6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.168ex; height:2.509ex;" alt="{\displaystyle a,b\in K}"> (<a href="/wiki/Kommutativgesetz" title="Kommutativgesetz">Kommutativgesetz</a>)</li> <li>Es gibt ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\in K\setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>∈</mo> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\in K\setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc64ed9091df8b40092c7aa2b12cfcf061eb62e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.751ex; height:2.843ex;" alt="{\displaystyle 1\in K\setminus \{0\}}">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\cdot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\cdot a=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c15b773211eeb31e5a62eabd1a03f4b4719f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.4ex; height:2.176ex;" alt="{\displaystyle 1\cdot a=a}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d838bfcfb39f7a33ffe31cd1c2a989b8ca3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle a\in K}"> (<a href="/wiki/Neutrales_Element" title="Neutrales Element">neutrales Element</a>).</li> <li>Zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in K\setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K\setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd82ae31888418a2bff3bdb89d275c1b72171e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.818ex; height:2.843ex;" alt="{\displaystyle a\in K\setminus \{0\}}"> existiert ein multiplikatives <a href="/wiki/Inverses_Element" title="Inverses Element">Inverses</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5709c8d86f7fec8fb86069bf5d15a9eabe564e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-1}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}\cdot a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}\cdot a=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13c9c7e33e30aebf1ff0bc1526350cec31bd0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.732ex; height:2.676ex;" alt="{\displaystyle a^{-1}\cdot a=1}">.</li></ol></li> <li>Zusammenspiel von additiver und multiplikativer Struktur: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8827e12f09f1ab8a5f3d7783b7357bd4cc398db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.324ex; height:2.843ex;" alt="{\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e030a1e5672b28c3b53d974ce2c4e72b03589ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.209ex; height:2.509ex;" alt="{\displaystyle a,b,c\in K}"> (Links-<a href="/wiki/Distributivgesetz" title="Distributivgesetz">Distributivgesetz</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abe5535680d9dbf6e1bd4b90284aa53addf4f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.324ex; height:2.843ex;" alt="{\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e030a1e5672b28c3b53d974ce2c4e72b03589ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.209ex; height:2.509ex;" alt="{\displaystyle a,b,c\in K}"> (Rechts-<a href="/wiki/Distributivgesetz" title="Distributivgesetz">Distributivgesetz</a>)</li></ol></li></ol> <dl><dd>      Aufgrund der multiplikativen Kommutativität würde es ausreichen, nur ein Distributivgesetz anzugeben.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Definition_als_spezieller_Ring">Definition als spezieller Ring</h3>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=4" title="Abschnitt bearbeiten: Definition als spezieller Ring" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Definition als spezieller Ring">Quelltext bearbeiten</a>]</div> Ein <a href="/wiki/Kommutativit%C3%A4t" class="mw-redirect" title="Kommutativität">kommutativer</a> <a href="/wiki/Ring_(Algebra)#Ring_mit_Eins_(unitärer_Ring)" title="Ring (Algebra)">unitärer Ring</a>, der nicht der <a href="/wiki/Nullring" title="Nullring">Nullring</a> ist, ist ein Körper, wenn in ihm jedes von Null verschiedene Element ein Inverses bezüglich der Multiplikation besitzt. Anders formuliert, ist ein Körper ein kommutativer unitärer Ring <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}">, in dem die <a href="/wiki/Einheitengruppe" title="Einheitengruppe">Einheitengruppe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee44d723b0d9bf440e04a664ea1e6e1958d743de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.148ex; height:2.343ex;" alt="{\displaystyle K^{*}}"> gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f12835c964f389ba3df759462067bf0087becac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.748ex; height:2.843ex;" alt="{\displaystyle K\setminus \{0\}}"> ist. <div class="mw-heading mw-heading3"><h3 id="Bemerkungen">Bemerkungen</h3>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=5" title="Abschnitt bearbeiten: Bemerkungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Bemerkungen">Quelltext bearbeiten</a>]</div> Die Definition sorgt dafür, dass in einem Körper in der „gewohnten“ Weise Addition, Subtraktion und Multiplikation funktionieren sowie die Division mit Ausnahme der <a href="/wiki/Division_(Mathematik)#Mathematischer_Beweis" title="Division (Mathematik)">nicht lösbaren Division durch 0</a>: <ul><li>Das Inverse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> bezüglich der Addition ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}"> und wird meist das additiv Inverse zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> oder auch das Negative von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> genannt.</li> <li>Das Inverse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> bezüglich der Multiplikation ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5709c8d86f7fec8fb86069bf5d15a9eabe564e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-1}}"> und wird das (multiplikativ) Inverse zu oder der Kehrwert von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> genannt.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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}"> ist das einzige Element des Körpers, das keinen Kehrwert hat, die <a href="/wiki/Multiplikative_Gruppe" class="mw-redirect" title="Multiplikative Gruppe">multiplikative Gruppe</a> eines Körpers ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{*}=K\setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{*}=K\setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba4ef0d2b8f4aeae8ea88713be6cffc4391acfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.995ex; height:2.843ex;" alt="{\displaystyle K^{*}=K\setminus \{0\}}">. Jegliche Lösung <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> jeder Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\cdot x=a\in K^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <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 0\cdot x=a\in K^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63b09e580fbc3d6759eb10122a1df559432bcc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.488ex; height:2.343ex;" alt="{\displaystyle 0\cdot x=a\in K^{*}}"> verletzt die Ringaxiome.</li></ul> Anmerkung: Die Bildung des Negativen eines Elementes hat nichts mit der Frage zu tun, ob das Element selbst negativ ist; beispielsweise ist das Negative der reellen Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e5b5b462e546b1d3d7e5f9a23efece405b2e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}"> die positive Zahl <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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}">. Allgemein gibt es in einem Körper keinen Begriff von negativen oder positiven Elementen. (Siehe auch <a href="/wiki/Geordneter_K%C3%B6rper" title="Geordneter Körper">geordneter Körper</a>.) <div class="mw-heading mw-heading3"><h3 id="Verallgemeinerungen:_Schiefkörper_und_Koordinatenkörper">Verallgemeinerungen: Schiefkörper und Koordinatenkörper</h3>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=6" title="Abschnitt bearbeiten: Verallgemeinerungen: Schiefkörper und Koordinatenkörper" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Verallgemeinerungen: Schiefkörper und Koordinatenkörper">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörper</a> und <a href="/wiki/Tern%C3%A4rk%C3%B6rper" title="Ternärkörper">Ternärkörper</a></div> Verzichtet man auf die Bedingung, dass die Multiplikation kommutativ ist, so gelangt man zur Struktur des Schiefkörpers. Es gibt jedoch auch Autoren, die bei einem Schiefkörper explizit voraussetzen, dass die Multiplikation nicht kommutativ ist. In diesem Fall sind die Begriffe Körper und Schiefkörper <a href="/wiki/Disjunkt" title="Disjunkt">disjunkt</a> – und nicht hierarchisch zueinander, wie sie es bei <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> sind, der <a href="/wiki/Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörper</a> als Körper und die hier besprochenen Körper als kommutative Körper bezeichnen. Ein Beispiel für einen echten Schiefkörper sind die <a href="/wiki/Quaternionen" class="mw-redirect" title="Quaternionen">Quaternionen</a>. In der <a href="/wiki/Analytische_Geometrie" title="Analytische Geometrie">analytischen Geometrie</a> werden Körper zur Koordinatendarstellung von Punkten in <a href="/wiki/Affiner_Raum" title="Affiner Raum">affinen</a> und <a href="/wiki/Projektiver_Raum" title="Projektiver Raum">projektiven Räumen</a> verwendet, siehe <a href="/wiki/Affine_Koordinaten" title="Affine Koordinaten">Affine Koordinaten</a>, <a href="/wiki/Projektives_Koordinatensystem" title="Projektives Koordinatensystem">Projektives Koordinatensystem</a>. In der <a href="/wiki/Synthetische_Geometrie" title="Synthetische Geometrie">synthetischen Geometrie</a>, in der auch Räume (insbesondere Ebenen) mit schwächeren Eigenschaften untersucht werden, benutzt man als Koordinatenbereiche („Koordinatenkörper“) auch Verallgemeinerungen der Schiefkörper, nämlich <a href="/wiki/Alternativk%C3%B6rper" title="Alternativkörper">Alternativkörper</a>, <a href="/wiki/Quasik%C3%B6rper" title="Quasikörper">Quasikörper</a> und <a href="/wiki/Tern%C3%A4rk%C3%B6rper" title="Ternärkörper">Ternärkörper</a>. <div class="mw-heading mw-heading2"><h2 id="Eigenschaften_und_Begriffe">Eigenschaften und Begriffe</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=7" title="Abschnitt bearbeiten: Eigenschaften und Begriffe" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Eigenschaften und Begriffe">Quelltext bearbeiten</a>]</div> <ul><li>Es gibt genau eine „0“ (Null-Element, <a href="/wiki/Neutrales_Element" title="Neutrales Element">neutrales Element</a> bzgl. der Körper-<a href="/wiki/Addition" title="Addition">Addition</a>) und eine „1“ (Eins-Element, neutrales Element bzgl. der Körper-<a href="/wiki/Multiplikation" title="Multiplikation">Multiplikation</a>) in einem Körper.</li> <li>Jeder Körper ist ein <a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">Ring</a>. Die Eigenschaften der multiplikativen Gruppe heben den Körper aus den Ringen heraus. Wenn die Kommutativität der multiplikativen Gruppe nicht gefordert wird, erhält man den Begriff des <a href="/wiki/Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörpers</a>.</li> <li>Jeder Körper ist <a href="/wiki/Nullteiler" title="Nullteiler">nullteilerfrei</a>: Ein Produkt zweier Elemente des Körpers ist genau dann 0, wenn mindestens einer der Faktoren 0 ist.</li> <li>Jedem Körper lässt sich eine <a href="/wiki/Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Charakteristik</a> zuordnen, die entweder 0 oder eine <a href="/wiki/Primzahl" title="Primzahl">Primzahl</a> ist.</li> <li>Die kleinste Teilmenge eines Körpers, die selbst noch alle Körperaxiome erfüllt, ist sein <a href="/wiki/Primk%C3%B6rper" title="Primkörper">Primkörper</a>. Der Primkörper ist entweder isomorph zum Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" 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} }"> der rationalen Zahlen (bei Körpern der Charakteristik 0) oder ein endlicher <a href="/wiki/Restklassenk%C3%B6rper" title="Restklassenkörper">Restklassenkörper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57869a3a3c4c431cc49c4c7ab1d9c7ea692b517b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.433ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> (bei Körpern der Charakteristik <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}">, speziell bei allen endlichen Körpern, s. u.).</li> <li>Ein Körper ist ein eindimensionaler Vektorraum über sich selbst als zugrundeliegendem Skalarkörper. Darüber hinaus existieren über allen Körpern Vektorräume beliebiger Dimension (siehe Hauptartikel <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a>).</li> <li>Ein wichtiges Mittel, um einen Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> algebraisch zu untersuchen, ist der <a href="/wiki/Polynomring" title="Polynomring">Polynomring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle K[X]}"> der <a href="/wiki/Polynom" title="Polynom">Polynome</a> in einer Variablen mit Koeffizienten aus <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}">. <ul><li>Man nennt einen Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> <a href="/wiki/Algebraisch_abgeschlossen" class="mw-redirect" title="Algebraisch abgeschlossen">algebraisch abgeschlossen</a>, wenn sich jedes nichtkonstante Polynom aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle K[X]}"> in Linearfaktoren aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle K[X]}"> zerlegen lässt.</li> <li>Man nennt einen Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> <a href="/wiki/K%C3%B6rpererweiterung#Vollkommen" title="Körpererweiterung">vollkommen</a>, wenn kein <a href="/wiki/Irreduzibles_Polynom" title="Irreduzibles Polynom">irreduzibles nichtkonstantes Polynom</a> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle K[X]}"> in irgendeiner Körpererweiterung mehrfache Nullstellen hat. Algebraische Abgeschlossenheit impliziert Vollkommenheit, aber nicht umgekehrt.</li></ul></li> <li>Wenn in einem Körper eine <a href="/wiki/Ordnungsrelation" title="Ordnungsrelation">Totalordnung</a> definiert ist, die mit der Addition und der Multiplikation verträglich ist, spricht man von einem <a href="/wiki/Geordneter_K%C3%B6rper" title="Geordneter Körper">geordneten Körper</a> und nennt die Totalordnung auch Anordnung des Körpers. In solchen Körpern kann man von negativen und positiven Zahlen sprechen. <ul><li>Wenn in dieser Anordnung jedes Körperelement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" 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 }"> durch eine endliche Summe des Einselementes übertroffen werden kann (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <1+1+\cdots +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <1+1+\cdots +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a27a04410dac72fe096c1763b4f5c54603bd78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.318ex; height:2.343ex;" alt="{\displaystyle \alpha <1+1+\cdots +1}">), sagt man, der Körper erfüllt das <a href="/wiki/Archimedisches_Axiom" title="Archimedisches Axiom">archimedische Axiom</a>, oder auch, er ist archimedisch geordnet.</li></ul></li> <li>In der <a href="/wiki/Bewertungstheorie" class="mw-redirect" title="Bewertungstheorie">Bewertungstheorie</a> werden bestimmte Körper mit Hilfe einer Bewertungsfunktion untersucht. Man nennt sie dann bewertete Körper.</li> <li>Ein Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> besitzt als Ring nur die trivialen <a href="/wiki/Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Ideale</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0)=\{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0)=\{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f05587dce3507a4ff35b954ec79c930514c0fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.558ex; height:2.843ex;" alt="{\displaystyle (0)=\{0\}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)=K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)=K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c46c59c337ca022efe484c07420e5cd021b597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.136ex; height:2.843ex;" alt="{\displaystyle (1)=K}">.</li> <li>Jeder nicht-konstante <a href="/wiki/Homomorphismus" title="Homomorphismus">Homomorphismus</a> von einem Körper in einen Ring ist <a href="/wiki/Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiv</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Körpererweiterung">Körpererweiterung</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=8" title="Abschnitt bearbeiten: Körpererweiterung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Körpererweiterung">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/K%C3%B6rpererweiterung" title="Körpererweiterung">Körpererweiterung</a></div> Eine Teilmenge <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> eines Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" 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}">, die selbst mit dessen Operationen wieder einen Körper bildet, wird Unter- oder Teilkörper genannt. Das Paar <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" 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}"> heißt Körpererweiterung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc1f8a493af0c60b015336bc212514b7d78c32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.747ex; height:2.176ex;" alt="{\displaystyle K\subset L}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0381945929156997b99bb43e5b7067d18c9a84b" 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}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L|K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L|K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3904a2706d41ef95ecff36f175ca971f0a116b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.296ex; height:2.843ex;" alt="{\displaystyle L|K}">. Beispielsweise ist der Körper der rationalen Zahlen <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" 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} }"> ein Teilkörper der reellen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }">. Eine Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}"> eines Körpers <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> ist ein Teilkörper, wenn sie folgende Eigenschaften hat: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{K}\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{K}\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0382dccde039402cebcacf4c1599fc002a57831d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.479ex; height:2.509ex;" alt="{\displaystyle 0_{K}\in U}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{K}\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{K}\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44163107d4970de2d2837c39b3a56409e50a32a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.479ex; height:2.509ex;" alt="{\displaystyle 1_{K}\in U}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in U\ \Rightarrow \ a+b\in U,\ a\cdot b\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>U</mi> <mtext> </mtext> <mo stretchy="false">⇒</mo> <mtext> </mtext> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>∈</mo> <mi>U</mi> <mo>,</mo> <mtext> </mtext> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in U\ \Rightarrow \ a+b\in U,\ a\cdot b\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfd1a28156263bc368520fa125f29a048aa7edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.495ex; height:2.509ex;" alt="{\displaystyle a,b\in U\ \Rightarrow \ a+b\in U,\ a\cdot b\in U}"> (<a href="/wiki/Abgeschlossenheit_(algebraische_Struktur)" title="Abgeschlossenheit (algebraische Struktur)">Abgeschlossenheit</a> bezüglich Addition und Multiplikation)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in U\ \Rightarrow \ -a\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>U</mi> <mtext> </mtext> <mo stretchy="false">⇒</mo> <mtext> </mtext> <mo>−</mo> <mi>a</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in U\ \Rightarrow \ -a\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab16c76e5cfdf2754d34ee51431c4ee4fb643a2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.322ex; height:2.343ex;" alt="{\displaystyle a\in U\ \Rightarrow \ -a\in U}"> (Zu jedem Element aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}"> ist auch das additive Inverse in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}">.)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in U\setminus \{0\}\ \Rightarrow \ a^{-1}\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>U</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo stretchy="false">⇒</mo> <mtext> </mtext> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in U\setminus \{0\}\ \Rightarrow \ a^{-1}\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec1d06a7fc16aba753654de4e4474ce797c62a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.496ex; height:3.176ex;" alt="{\displaystyle a\in U\setminus \{0\}\ \Rightarrow \ a^{-1}\in U}"> (Zu jedem Element aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}"> mit Ausnahme der Null ist auch das multiplikativ Inverse in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" 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}">.)</li></ul> Das algebraische Teilgebiet, das sich mit der Untersuchung von Körpererweiterungen beschäftigt, ist die <a href="/wiki/Galoistheorie" title="Galoistheorie">Galoistheorie</a>. <div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=9" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Beispiele">Quelltext bearbeiten</a>]</div> <ul><li>Bekannte Beispiele für Körper sind <ul><li>der Körper der <a href="/wiki/Rationale_Zahlen" class="mw-redirect" title="Rationale Zahlen">rationalen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} ,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ,+,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b9495c5986da1b645da13cf5421c6fc06abcc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.14ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} ,+,\cdot )}">, d. h. die <a href="/wiki/Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> der <a href="/wiki/Rationale_Zahlen" class="mw-redirect" title="Rationale Zahlen">rationalen Zahlen</a> mit der üblichen <a href="/wiki/Addition" title="Addition">Addition</a> und <a href="/wiki/Multiplikation" title="Multiplikation">Multiplikation</a></li> <li>der Körper der <a href="/wiki/Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reellen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,+,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed3304e521dac824c9b33476be16be5fadde282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.01ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+,\cdot )}">, d. h. die Menge der reellen Zahlen mit der üblichen Addition und Multiplikation, und</li> <li>der Körper der <a href="/wiki/Komplexe_Zahlen" class="mw-redirect" title="Komplexe Zahlen">komplexen Zahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {C} ,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {C} ,+,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c13ca9956454acbfcbe7e264150781d824b0ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.01ex; height:2.843ex;" alt="{\displaystyle (\mathbb {C} ,+,\cdot )}"> d. h. die Menge der komplexen Zahlen mit der üblichen Addition und Multiplikation.</li></ul></li></ul> <ul><li>Körper können durch <a href="/wiki/Adjunktion_(Algebra)" title="Adjunktion (Algebra)">Adjunktion</a> erweitert werden. Ein wichtiger Spezialfall – insbesondere in der <a href="/wiki/Galoistheorie" title="Galoistheorie">Galoistheorie</a> – sind <a href="/wiki/Algebraische_Erweiterung" title="Algebraische Erweiterung">algebraische Körpererweiterungen</a> des Körpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74881d420116e547703edb7942c6448853d6ebd" 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 \textstyle \mathbb {Q} }">. Der Erweiterungskörper kann dabei als <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraum</a> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74881d420116e547703edb7942c6448853d6ebd" 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 \textstyle \mathbb {Q} }"> aufgefasst werden. <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})=\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>∣</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})=\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0e49215a504ec98fbba51676bada5a33996b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.152ex; height:3.176ex;" alt="{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})=\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \}}"> ist ein Körper. Es genügt zu zeigen, dass das Inverse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle a+b{\sqrt {2}}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle a+b{\sqrt {2}}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1091ebdd1dc84ab6b95bfd48a86fcb248dea528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.427ex; height:3.176ex;" alt="{\displaystyle \textstyle a+b{\sqrt {2}}\neq 0}"> auch von der angegebenen Form ist:       <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{a+b{\sqrt {2}}}}={\frac {(a-b{\sqrt {2}})}{(a+b{\sqrt {2}})\cdot (a-b{\sqrt {2}})}}={\frac {(a-b{\sqrt {2}})}{(a^{2}-2b^{2})}}={\frac {a}{(a^{2}-2b^{2})}}+{\frac {-b}{(a^{2}-2b^{2})}}{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{a+b{\sqrt {2}}}}={\frac {(a-b{\sqrt {2}})}{(a+b{\sqrt {2}})\cdot (a-b{\sqrt {2}})}}={\frac {(a-b{\sqrt {2}})}{(a^{2}-2b^{2})}}={\frac {a}{(a^{2}-2b^{2})}}+{\frac {-b}{(a^{2}-2b^{2})}}{\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf3dd3687860a7be77be414f76343638db4aaef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:79.655ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{a+b{\sqrt {2}}}}={\frac {(a-b{\sqrt {2}})}{(a+b{\sqrt {2}})\cdot (a-b{\sqrt {2}})}}={\frac {(a-b{\sqrt {2}})}{(a^{2}-2b^{2})}}={\frac {a}{(a^{2}-2b^{2})}}+{\frac {-b}{(a^{2}-2b^{2})}}{\sqrt {2}}}"> Eine mögliche Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a5fa9c06f803f40982fce97662b36e2ae24105" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:3.176ex;" alt="{\displaystyle \textstyle \mathbb {Q} ({\sqrt {2}})}"> ist {<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1,{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle 1,{\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0683242686573b7ff20b7aa0d2a24ea4436a10b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.295ex; height:3.009ex;" alt="{\displaystyle \textstyle 1,{\sqrt {2}}}">}.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>∣</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f4b37dfa05b4fc926f31644beb2ba507bfa810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.769ex; height:3.343ex;" alt="{\displaystyle \mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\}}"> ist ein Körper mit Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{1,{\sqrt {2}},{\sqrt {3}},{\sqrt {6}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{1,{\sqrt {2}},{\sqrt {3}},{\sqrt {6}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c311be9e72b2f5f8cfbb31d2ece5f413f57d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.27ex; height:3.343ex;" alt="{\displaystyle \left\{1,{\sqrt {2}},{\sqrt {3}},{\sqrt {6}}\right\}}">.</li></ul></li></ul> <ul><li>Weitere Beispiele liefern die <a href="/wiki/Restklassenk%C3%B6rper" title="Restklassenkörper">Restklassenkörper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db14ec2f239766eda9dfc1f364dd45b1230169c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.01ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}"> mit <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> <a href="/wiki/Primzahl" title="Primzahl">Primzahl</a><a href="#cite_note-Beutelspacher-LA-7-35-1">[1]</a> und <ul><li>deren endliche <a href="/wiki/K%C3%B6rpererweiterung" title="Körpererweiterung">Körpererweiterungen</a>, die <a href="/wiki/Endlicher_K%C3%B6rper" title="Endlicher Körper">endlichen Körper</a>,</li> <li>allgemeiner deren algebraische Körpererweiterungen, die <a href="/wiki/Frobeniushomomorphismus" title="Frobeniushomomorphismus">Frobeniuskörper</a>, und</li> <li>noch allgemeiner deren beliebige Körpererweiterungen, die Körper mit <a href="/wiki/Charakteristik_(Algebra)" title="Charakteristik (Algebra)">Primzahlcharakteristik</a>.</li></ul></li> <li>Zu jeder Primzahl <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> der Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" 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}}"> der <a href="/wiki/P-adische_Zahlen" class="mw-redirect" title="P-adische Zahlen">p-adischen Zahlen</a>.</li> <li>Die Menge der ganzen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} ,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,+,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a626bb1acb2b3beebb7ed25b98f0cc7fdb7df60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.882ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,+,\cdot )}"> mit den üblichen Verknüpfungen ist kein Körper: Zwar ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910eaae0a8267ccb04d4846f6a28f02ce6ab8ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.202ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,+)}"> eine Gruppe mit neutralem Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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}"> und jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175ebd03e0644b0967a63d648c2843a5e883257b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.621ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {Z} }"> besitzt das additive Inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}">, aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} \setminus \{0\},\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} \setminus \{0\},\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c2cb62727866a32ac199dc3419a9d5ffa393fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.722ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} \setminus \{0\},\cdot )}"> ist keine Gruppe. Immerhin ist <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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}"> das neutrale Element, aber außer zu <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> gibt es keine multiplikativen Inversen (zum Beispiel ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{-1}=1/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{-1}=1/3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4226d8b4a581e9348516b12a9d0d4715f7d49347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.081ex; height:3.176ex;" alt="{\displaystyle 3^{-1}=1/3}"> keine ganze, sondern eine echt rationale Zahl): <ul><li>Die ganzen Zahlen bilden lediglich einen <a href="/wiki/Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a>, dessen <a href="/wiki/Quotientenk%C3%B6rper" title="Quotientenkörper">Quotientenkörper</a> die rationalen Zahlen sind.</li></ul></li> <li>Das Konzept, mit dem sich der Integritätsring der ganzen Zahlen zum Körper der rationalen Zahlen erweitern und in diesen einbetten lässt, kann auf beliebige Integritätsringe verallgemeinert werden: <ul><li>So entsteht in der <a href="/wiki/Funktionentheorie" title="Funktionentheorie">Funktionentheorie</a> aus dem Integritätsring der auf einem <a href="/wiki/Gebiet_(Mathematik)" title="Gebiet (Mathematik)">Gebiet</a> der komplexen Zahlenebene <a href="/wiki/Holomorphie" class="mw-redirect" title="Holomorphie">holomorphen Funktionen</a> der Körper der auf demselben Gebiet <a href="/wiki/Meromorphe_Funktion" title="Meromorphe Funktion">meromorphen Funktionen</a>, und abstrakter</li> <li>aus dem Integritätsring der formalen <a href="/wiki/Potenzreihe" title="Potenzreihe">Potenzreihen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[[x]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[[x]]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/360aa2d367ac90bc66835442e10c739356f67dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.983ex; height:2.843ex;" alt="{\displaystyle K[[x]]}"> über einem Körper <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> dessen Quotientenkörper, analog aus dem Integritätsring der formalen <a href="/wiki/Dirichletreihe" title="Dirichletreihe">Dirichletreihen</a>,</li> <li>aus dem Ring der <a href="/wiki/Polynom" title="Polynom">Polynome</a> in <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"> Variablen, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[x_{1},x_{2},\dots ,x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[x_{1},x_{2},\dots ,x_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf50d57cd977cb0e672d455dd348ea18d7ecdf24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.888ex; height:2.843ex;" alt="{\displaystyle K[x_{1},x_{2},\dots ,x_{n}]}">, dessen Quotientenkörper, der Körper der rationalen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(x_{1},x_{2},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x_{1},x_{2},\dots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d20a59f6420e88dc090a86678bd0e37be2eda07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.403ex; height:2.843ex;" alt="{\displaystyle K(x_{1},x_{2},\dots ,x_{n})}"> in ebenso vielen Variablen.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Endliche_Körper">Endliche Körper</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=10" title="Abschnitt bearbeiten: Endliche Körper" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Endliche Körper">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Endlicher_K%C3%B6rper" title="Endlicher Körper">Endlicher Körper</a></div> Ein Körper ist ein endlicher Körper, wenn seine Grundmenge <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> endlich ist. Die endlichen Körper sind in folgendem Sinne vollständig klassifiziert: Jeder endliche Körper hat genau <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b95f6525025c9ae1432ea9bd50c3c1cab4b7c6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.556ex; height:2.676ex;" alt="{\displaystyle q=p^{n}}"> Elemente mit einer Primzahl <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> und einer positiven <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahl</a> <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}">. Bis auf <a href="/wiki/Isomorphie_(Mathematik)" class="mw-redirect" title="Isomorphie (Mathematik)">Isomorphie</a> gibt es zu jedem solchen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> genau einen endlichen Körper, der mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}"> <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>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb96e056c071d13fc7702013f9273e7f5cd88a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}"> bezeichnet wird. Jeder Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p^{n}}}"> <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"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ef6f7b38b948ec1d7d54dfa483c00db7dedd5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.444ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p^{n}}}"> hat die Charakteristik <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}">. Im Artikel <a href="/wiki/Endlicher_K%C3%B6rper#Der_Körper_mit_4_Elementen" title="Endlicher Körper">Endlicher Körper</a> werden die Additions- und Multiplikationstafeln des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{4}}"> <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"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e70c988f7702e675b4ef121c3c68738956c83f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{4}}"> gezeigt bei farbiger Hervorhebung von dessen Unterkörper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}}"> <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"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde97f1e971e76227cd0aac645b7b0901d7b668d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \mathbb {F} _{2}}">. Im Spezialfall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"> erhalten wir zu jeder Primzahl <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> den Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35035371db7bee93733c68c1802114c17d8bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.479ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p}}">, der isomorph ist zum <a href="/wiki/Restklassenk%C3%B6rper" title="Restklassenkörper">Restklassenkörper</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57869a3a3c4c431cc49c4c7ab1d9c7ea692b517b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.433ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> und <a href="/wiki/Primk%C3%B6rper" title="Primkörper">Primkörper</a> der (Primzahl)charakteristik <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> genannt wird. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" 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}"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p^{n}}}"> <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"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ef6f7b38b948ec1d7d54dfa483c00db7dedd5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.444ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p^{n}}}"> niemals isomorph zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f568eec3d80f812c3d02a19a445f088e888c180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.651ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }">; stattdessen ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p^{n}}}"> <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"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ef6f7b38b948ec1d7d54dfa483c00db7dedd5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.444ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{p^{n}}}"> isomorph zu <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /p\mathbb {Z} )[X]/(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /p\mathbb {Z} )[X]/(P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/208bc8d71fdeb61297b34fafbc8dd64ba45add20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.233ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /p\mathbb {Z} )[X]/(P)}">,</dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb4d802ca5718a14dc961af8692f35cdfad169b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle K[X]}"> den Ring der <a href="/wiki/Polynom" title="Polynom">Polynome</a> mit Koeffizienten in <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" 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}"> darstellt (hier ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Z} /p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Z} /p\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d99602402dcfb9098b48b855a1d79b50e8f507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.597ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {Z} /p\mathbb {Z} }">) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\in (\mathbb {Z} /p\mathbb {Z} )[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\in (\mathbb {Z} /p\mathbb {Z} )[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657ddb5c6f7e34468e69ef026685e1a99dd529d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.102ex; height:2.843ex;" alt="{\displaystyle P\in (\mathbb {Z} /p\mathbb {Z} )[X]}"> ein <a href="/wiki/Irreduzibles_Element" class="mw-redirect" title="Irreduzibles Element">irreduzibles</a> Polynom vom Grad <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"> ist. In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /p\mathbb {Z} )[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /p\mathbb {Z} )[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67499db475fa009dfd134de69c9ea7c1ff033a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.516ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /p\mathbb {Z} )[X]}"> ist ein Polynom irreduzibel, wenn aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=P_{1}\cdot P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=P_{1}\cdot P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf775757da85121ac0fe74019fac9cff821840f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.616ex; height:2.509ex;" alt="{\displaystyle P=P_{1}\cdot P_{2}}"> folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" 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_{1}}"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" 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_{2}}"> ein Element von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57869a3a3c4c431cc49c4c7ab1d9c7ea692b517b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.433ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> ist, also ein konstantes Polynom. Hier bedeutet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"> das von <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"> <a href="/wiki/Ideal_(Ringtheorie)#Erzeugung_von_Idealen" title="Ideal (Ringtheorie)">erzeugte Ideal</a>. <div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=11" title="Abschnitt bearbeiten: Geschichte" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Geschichte">Quelltext bearbeiten</a>]</div> Wesentliche Ergebnisse der Körpertheorie sind <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> und <a href="/wiki/Ernst_Steinitz" title="Ernst Steinitz">Ernst Steinitz</a> zu verdanken. Weitere Einzelheiten zur Genese des Begriffes liefert <a href="/wiki/Wulf-Dieter_Geyer" title="Wulf-Dieter Geyer">Wulf-Dieter Geyer</a> in Kapitel 2 seines Beitrages, in dem er u. a. auf die Rolle <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekinds</a> hinweist (siehe <a class="mw-selflink-fragment" href="#Literatur">Literatur</a>). <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=12" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Siehe auch">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper">Algebraischer Zahlkörper</a></li> <li><a href="/wiki/Ring_(Algebra)" title="Ring (Algebra)">Ring (Algebra)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=13" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: Algebra. 7. Auflage. Springer-Verlag, 2009, <a href="/wiki/Spezial:ISBN-Suche/3540403884" class="internal mw-magiclink-isbn">ISBN 3-540-40388-4</a>, <a href="//doi.org/10.1007/978-3-540-92812-6" class="extiw" title="doi:10.1007/978-3-540-92812-6">doi:10.1007/978-3-540-92812-6</a>.</li> <li><a href="/w/index.php?title=Thomas_W._Hungerford&action=edit&redlink=1" class="new" title="Thomas W. Hungerford (Seite nicht vorhanden)">Thomas W. Hungerford</a>: Algebra. 5. Auflage. Springer-Verlag, 1989, <a href="/wiki/Spezial:ISBN-Suche/0387905189" class="internal mw-magiclink-isbn">ISBN 0-387-90518-9</a>.</li> <li><a href="/wiki/Wulf-Dieter_Geyer" title="Wulf-Dieter Geyer">Wulf-Dieter Geyer</a>: <a rel="nofollow" class="external text" href="https://wwwfr.uni.lu/content/download/75426/940966/file/">Field Theory.</a> In: Volume I of the Proceedings of the Qinter School on Galois Theory, 15-24 February 2012, Université du Luxembourg, Luxembourg. Juli 2013, abgerufen am 9. November 2022.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AK%C3%B6rper+%28Algebra%29&rft.title=Field+Theory&rft.description=Field+Theory&rft.identifier=https%3A%2F%2Fwwwfr.uni.lu%2Fcontent%2Fdownload%2F75426%2F940966%2Ffile%2F&rft.creator=%5B%5BWulf-Dieter+Geyer%5D%5D&rft.date=2013-07">  siehe insbesondere Kapitel 2 („Historical remarks about the concept of field“), Seite 29.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=14" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></div><a href="https://de.wikibooks.org/wiki/Mathe_f%C3%BCr_Nicht-Freaks:_K%C3%B6rperaxiome" class="extiw" title="b:Mathe für Nicht-Freaks: Körperaxiome">Wikibooks: Mathe für Nicht-Freaks: Körperaxiome</a> – Lern- und Lehrmaterialien</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></div><a href="https://de.wikibooks.org/wiki/Mathe_f%C3%BCr_Nicht-Freaks:_Folgerungen_aus_den_K%C3%B6rperaxiomen" class="extiw" title="b:Mathe für Nicht-Freaks: Folgerungen aus den Körperaxiomen">Wikibooks: Mathe für Nicht-Freaks: Folgerungen aus den Körperaxiomen</a> – Lern- und Lehrmaterialien</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /><a href="https://de.wiktionary.org/wiki/K%C3%B6rper" class="extiw" title="wikt:Körper">Wiktionary: Körper</a> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=K%C3%B6rper_(Algebra)&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=K%C3%B6rper_(Algebra)&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-Beutelspacher-LA-7-35-1"><a href="#cite_ref-Beutelspacher-LA-7-35_1-0">↑</a> <a href="/wiki/Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>: <cite style="font-style:italic">Lineare Algebra</cite>. 7. Auflage. <a href="/wiki/Vieweg%2BTeubner_Verlag" class="mw-redirect" title="Vieweg+Teubner Verlag">Vieweg+Teubner Verlag</a>, Wiesbaden 2010, <a href="/wiki/Spezial:ISBN-Suche/9783528665081" class="internal mw-magiclink-isbn">ISBN 978-3-528-66508-1</a>, S. 35–37.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:K%C3%B6rper+%28Algebra%29&rft.au=Albrecht+Beutelspacher&rft.btitle=Lineare+Algebra&rft.date=2010&rft.edition=7&rft.genre=book&rft.isbn=9783528665081&rft.pages=35-37&rft.place=Wiesbaden&rft.pub=Vieweg%2BTeubner+Verlag" style="display:none">  </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Körper_(Algebra)&oldid=244759049">https://de.wikipedia.org/w/index.php?title=Körper_(Algebra)&oldid=244759049</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:K%C3%B6rper_(Algebra)" title="Kategorie:Körper (Algebra)">Körper (Algebra)</a></li><li><a href="/wiki/Kategorie:K%C3%B6rpertheorie" title="Kategorie:Körpertheorie">Körpertheorie</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=K%C3%B6rper+%28Algebra%29" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tallkropp" title="Tallkropp – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Tallkropp" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cia%C5%82o_(matematyka)" title="Ciało (matematyka) – Polnisch" lang="pl" hreflang="pl" data-title="Ciało (matematyka)" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Camp_(matem%C3%A0tica)" title="Camp (matemàtica) – Piemontesisch" lang="pms" hreflang="pms" data-title="Camp (matemàtica)" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática) – Portugiesisch" lang="pt" hreflang="pt" data-title="Corpo (matemática)" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Corp_comutativ" title="Corp comutativ – Rumänisch" lang="ro" hreflang="ro" data-title="Corp comutativ" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Поле (алгебра) – Russisch" lang="ru" hreflang="ru" data-title="Поле (алгебра)" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Campu_(matimatica)" title="Campu (matimatica) – Sizilianisch" lang="scn" hreflang="scn" data-title="Campu (matimatica)" data-language-autonym="Sicilianu" data-language-local-name="Sizilianisch" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Polje_(matematika)" title="Polje (matematika) – Serbokroatisch" lang="sh" hreflang="sh" data-title="Polje (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a 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