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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Noetherscher Normalisierungssatz</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>noethersche Normalisierungssatz</b> (oder auch <i>noethersches Normalisierungslemma</i>) (nach <a href="Emmy_Noether" title="Emmy Noether">Emmy Noether</a>) ist eine Strukturaussage aus dem <a href="Mathematik" title="Mathematik">mathematischen</a> Teilgebiet der <a href="Kommutative_Algebra" title="Kommutative Algebra">kommutativen Algebra</a>. In geometrischer Sprache besagt er, dass es von einem geometrischen Objekt stets eine Abbildung in einen affinen Raum gibt, deren Fasern endlich sind.
</p><p><i>Dieser Artikel beschäftigt sich mit kommutativer Algebra. Insbesondere sind alle betrachteten Ringe kommutativ und haben ein Einselement. Für weitere Details siehe <a href="Kommutative_Algebra" title="Kommutative Algebra">Kommutative Algebra</a>.</i>
</p>
<div class="mw-heading mw-heading2"><h2 id="Formulierung">Formulierung</h2></div>
<p>Es sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> ein <a href="K%C3%B6rpertheorie" class="mw-redirect" title="Körpertheorie">Körper</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> eine <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-<a href="Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebra</a> <a href="Endlichkeitsbedingungen_der_algebraischen_Geometrie" title="Endlichkeitsbedingungen der algebraischen Geometrie">endlichen Typs</a>. Dann gibt es algebraisch unabhängige Elemente <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}\in A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}\in A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a80dddbe85659616ba3c7f034548f4480f13ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.694ex; height:2.509ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}\in A}" loading="lazy"></span>, so dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> eine <a href="Endlichkeitsbedingungen_der_algebraischen_Geometrie" title="Endlichkeitsbedingungen der algebraischen Geometrie">endliche</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{1},\ldots ,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>
<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},\ldots ,x_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37c2b680cd4b215ac5c3c548a0e596d534526cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.615ex; height:2.843ex;" alt="{\displaystyle k[x_{1},\ldots ,x_{n}]}" loading="lazy"></span>-Algebra, also <a href="Ganzes_Element#Definition" title="Ganzes Element">ganz über</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{1},\ldots ,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>
<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},\ldots ,x_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37c2b680cd4b215ac5c3c548a0e596d534526cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.615ex; height:2.843ex;" alt="{\displaystyle k[x_{1},\ldots ,x_{n}]}" loading="lazy"></span> ist. Man kann für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> den <a href="Transzendenzgrad" class="mw-redirect" title="Transzendenzgrad">Transzendenzgrad</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Trg} (A\colon k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Trg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>:<!-- : --></mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Trg} (A\colon k)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3754104e8b2a627bcb05f0ab707660744249541f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.55ex; height:2.843ex;" alt="{\displaystyle \operatorname {Trg} (A\colon k)}" loading="lazy"></span> wählen.
</p><p>Dabei bedeutet „algebraisch unabhängig“, dass der Homomorphismus
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[X_{1},\ldots ,X_{n}]\to A,\quad X_{i}\mapsto x_{i}}">
<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>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>A</mi>
<mo>,</mo>
<mspace width="1em"></mspace>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">↦<!-- ↦ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k[X_{1},\ldots ,X_{n}]\to A,\quad X_{i}\mapsto x_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7061f97ce64b2336d29caacd36321736d6f814f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.985ex; height:2.843ex;" alt="{\displaystyle k[X_{1},\ldots ,X_{n}]\to A,\quad X_{i}\mapsto x_{i}}" loading="lazy"></span></dd></dl>
<p>aus dem Polynomring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[X_{1},\ldots ,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>
<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},\ldots ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff472a050c05d0bbc73bd23888d3d654df28fbfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.804ex; height:2.843ex;" alt="{\displaystyle k[X_{1},\ldots ,X_{n}]}" loading="lazy"></span> nach <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> injektiv ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Ganzheit_(kommutative_Algebra)" class="mw-redirect" title="Ganzheit (kommutative Algebra)">Ganzheit (kommutative Algebra)</a></li>
<li><a href="Hilbertscher_Nullstellensatz" title="Hilbertscher Nullstellensatz">Hilbertscher Nullstellensatz</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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