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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Primitives Polynom</span></h1>
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<p>In der Theorie <a href="Mathematik" title="Mathematik">mathematischer</a> <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> ist ein <b>primitives Polynom</b> das <a href="Minimalpolynom_(K%C3%B6rpertheorie)" title="Minimalpolynom (Körpertheorie)">Minimalpolynom</a> einer primitiven <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p^{m}-1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (p^{m}-1)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5fc37622e19bd1d759658d8473eda2005225b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.657ex; height:2.843ex;" alt="{\displaystyle (p^{m}-1)}" loading="lazy"></span>-ten <a href="Einheitswurzel" title="Einheitswurzel">Einheitswurzel</a> einer <a href="K%C3%B6rpererweiterung" title="Körpererweiterung">Körpererweiterung</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 \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p^{m})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10b4cca67765531b15e118a318002ba763580ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.996ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p^{m})}" loading="lazy"></span> über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p)}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dcb6720c421397627afb8ecce23a98b152d432f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.321ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p)}" loading="lazy"></span> <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">endlicher Körper</a>.
Anders ausgedrückt ist ein Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> mit den Koeffizienten aus <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 \mathrm {GF} (p)=\mathbb {Z} /p\mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<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 \mathrm {GF} (p)=\mathbb {Z} /p\mathbb {Z} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a37a850a2372db11b72bd68c3bc4f10f7849a23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.852ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p)=\mathbb {Z} /p\mathbb {Z} }" loading="lazy"></span> ein primitives Polynom, wenn es eine Nullstelle <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 \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> in <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 \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p^{m})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10b4cca67765531b15e118a318002ba763580ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.996ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p^{m})}" loading="lazy"></span> hat, so dass die Menge <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 \{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\dots ,\alpha ^{p^{m}-2}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mi>α<!-- α --></mi>
<mo>,</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msup>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\dots ,\alpha ^{p^{m}-2}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53b4078c321d9e5faed620b343c087ccfd9b0b8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.518ex; height:3.176ex;" alt="{\displaystyle \{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\dots ,\alpha ^{p^{m}-2}\}}" loading="lazy"></span> der ganze Körper <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 \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p^{m})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10b4cca67765531b15e118a318002ba763580ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.996ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p^{m})}" loading="lazy"></span> ist und außerdem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> das Polynom mit dem kleinsten Grad mit <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 \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> als <a href="Nullstelle" title="Nullstelle">Nullstelle</a> ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<p>Da alle Minimalpolynome <a href="Irreduzibles_Polynom" title="Irreduzibles Polynom">irreduzibel</a> sind, sind primitive Polynome ebenso irreduzibel.
</p><p>Ein primitives Polynom muss einen von Null verschiedenen konstanten Term haben, da es andernfalls durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> teilbar wäre. Über einem Körper aus zwei Elementen ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle X+1}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c604b86dfea83bda80bc69a22e046147948115.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.983ex; height:2.343ex;" alt="{\displaystyle X+1}" loading="lazy"></span> ein primitives Polynom und alle anderen primitiven Polynome haben eine ungerade Anzahl von Termen, da jedes Polynom modulo 2 mit einer geraden Anzahl von Termen durch <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x+1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16df430ed7a23df9b160a5bbd957f306a0c3baa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x+1}" loading="lazy"></span> teilbar ist.
</p><p>Ein irreduzibles Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> des Grades <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 m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dcb6720c421397627afb8ecce23a98b152d432f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.321ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p)}" loading="lazy"></span> für eine <a href="Primzahl" title="Primzahl">Primzahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> ist ein primitives Polynom, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{m}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{m}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3051bc4eef2dd66102f9c5e5ba70e439acef60b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.937ex; height:2.676ex;" alt="{\displaystyle p^{m}-1}" loading="lazy"></span> die kleinste ganze Zahl <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> ist, für die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> ein Teiler von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{n}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f48db34e8e99d3b3d6d19d31f481a74b8de1ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.218ex; height:2.509ex;" alt="{\displaystyle X^{n}-1}" loading="lazy"></span> ist.
</p><p>Über dem Körper <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 \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p^{m})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10b4cca67765531b15e118a318002ba763580ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.996ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p^{m})}" loading="lazy"></span> gibt es genau <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 {\tfrac {\varphi (p^{m}-1)}{m}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mi>m</mi>
</mfrac>
</mstyle>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\varphi (p^{m}-1)}{m}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07f913212d60f77233f34aa44f7d52be0b4f0634.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.453ex; height:4.009ex;" alt="{\displaystyle {\tfrac {\varphi (p^{m}-1)}{m}}}" loading="lazy"></span> primitive Polynome des Grades <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 m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>, wobei <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 \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> die <a href="Eulersche_%CF%86-Funktion" class="mw-redirect" title="Eulersche φ-Funktion">Eulersche φ-Funktion</a> ist.
</p><p>Die Nullstellen eines primitiven Polynoms haben alle die Ordnung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{m}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{m}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3051bc4eef2dd66102f9c5e5ba70e439acef60b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.937ex; height:2.676ex;" alt="{\displaystyle p^{m}-1}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Darstellung_von_Körper-Elementen"><span id="Darstellung_von_K.C3.B6rper-Elementen"></span>Darstellung von Körper-Elementen</h3></div>
<p>Primitive Polynome werden für die Darstellung von Elementen eines <a href="Endlicher_K%C3%B6rper" title="Endlicher Körper">endlichen Körpers</a> verwendet. Wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathrm {GF} (p^{m})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70349f1044cc8d24f8a9d345e915f9a6b6a871c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.324ex; height:2.843ex;" alt="{\displaystyle \alpha \in \mathrm {GF} (p^{m})}" loading="lazy"></span> eine Nullstelle eines primitiven Polynoms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> ist, dann hat <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 \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> die Ordnung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{m}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{m}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3051bc4eef2dd66102f9c5e5ba70e439acef60b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.937ex; height:2.676ex;" alt="{\displaystyle p^{m}-1}" loading="lazy"></span>, das heißt alle Elemente von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p^{m})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c10b4cca67765531b15e118a318002ba763580ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.996ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p^{m})}" loading="lazy"></span> können als aufeinanderfolgende Potenzen von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> dargestellt werden:
</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 \mathrm {GF} (p^{m})=\{0,1,\alpha ,\alpha ^{2},\ldots ,\alpha ^{p^{m}-2}\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mi>α<!-- α --></mi>
<mo>,</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msup>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msup>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p^{m})=\{0,1,\alpha ,\alpha ^{2},\ldots ,\alpha ^{p^{m}-2}\}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2599ef47deef27468990ed543ec11d939ac21c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.683ex; height:3.176ex;" alt="{\displaystyle \mathrm {GF} (p^{m})=\{0,1,\alpha ,\alpha ^{2},\ldots ,\alpha ^{p^{m}-2}\}.}" loading="lazy"></span></dd></dl>
<p>Wenn diese Elemente modulo <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> reduziert werden, dann bildet die Darstellung als polynomielle Basis aller dieser Elemente einen Körper.
</p><p>Da die <a href="Multiplikative_Gruppe" class="mw-redirect" title="Multiplikative Gruppe">multiplikative Gruppe</a> eines endlichen Körpers immer eine <a href="Zyklische_Gruppe" title="Zyklische Gruppe">zyklische Gruppe</a> ist, ist für ein primitives Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00816772e8dff4e6733c478ec77fab0382264a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}" loading="lazy"></span> das Element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ein Generator der multiplikativen Gruppe in <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 \mathrm {GF} (p)[X]/(F(X))\cong \mathrm {GF} (p^{m})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<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>F</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>≅<!-- ≅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {GF} (p)[X]/(F(X))\cong \mathrm {GF} (p^{m})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4ecab5b9dcd6512d13b60700dbe42259e61c10f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.191ex; height:2.843ex;" alt="{\displaystyle \mathrm {GF} (p)[X]/(F(X))\cong \mathrm {GF} (p^{m})}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Erzeugung_von_Pseudo-Zufallszahlen">Erzeugung von Pseudo-Zufallszahlen</h3></div>
<p>Primitive Polynome definieren eine wiederkehrende Relation, die verwendet werden kann, um Bits von <a href="Pseudozufallszahl" class="mw-redirect" title="Pseudozufallszahl">Pseudozufallszahlen</a> zu erzeugen. Tatsächlich steht jedes <a href="Linear_r%C3%BCckgekoppeltes_Schieberegister" title="Linear rückgekoppeltes Schieberegister">linear rückgekoppelte Schieberegister</a> mit maximalem Zyklus (dieser ist 2<sup>lrsr length</sup> - 1) mit primitiven Polynomen in Beziehung.
</p><p>Sei z. B. ein primitives Polynom <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^{10}+X^{3}+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{10}+X^{3}+1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd4dae673b24af7bf0390cd65100c4116fe09600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.767ex; height:2.843ex;" alt="{\displaystyle X^{10}+X^{3}+1}" loading="lazy"></span> gegeben. Man beginnt mit einem benutzerdefinierten Startwert (engl. <i>seed</i>, <i>Saatkorn</i>, dieser muss nicht unbedingt zufällig gewählt werden). Man nimmt dann das 10-te, 3-te und 0-te <a href="Bit" title="Bit">Bit</a>, gezählt vom niederwertigsten Bit, verknüpft diese mit <a href="Kontravalenz" title="Kontravalenz">XOR</a> und erhält ein neues Bit. Die Saatzahl wird dann nach links verschoben und das neue Bit wird zum niederwertigsten Bit der Saatzahl. Dies kann wiederholt werden um <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{10}-1=1023}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>10</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>=</mo>
<mn>1023</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{10}-1=1023}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/090162ce9f802b48e2cf396aff9d51e6918874e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.79ex; height:2.843ex;" alt="{\displaystyle 2^{10}-1=1023}" loading="lazy"></span> Pseudo-Zufalls-Bits zu erzeugen. Für eine <a href="Maximum_Length_Sequence" title="Maximum Length Sequence">Maximum Length Sequence</a> sind ganz bestimmte Ausgänge des Schieberegisters erforderlich.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Allgemein gilt für ein primitives Polynom des Grades <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 m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>, dass dieser Vorgang <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{m}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{m}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79530a5fede656dda33fb3890829a22821d79174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.84ex; height:2.509ex;" alt="{\displaystyle 2^{m}-1}" loading="lazy"></span> Pseudo-Zufallszahlen erzeugt, bevor die Sequenz sich wiederholt.
</p>
<div class="mw-heading mw-heading3"><h3 id="Primitive_Trinome">Primitive Trinome</h3></div>
<p>Primitive <a href="Trinom" title="Trinom">Trinome</a> sind primitive Polynome mit nur drei von Null verschiedenen Termen. Die Trinome sind sehr einfach und werden für sehr effiziente Zufallszahlengeneratoren verwendet. Es gibt verschiedene Methoden, um primitive Trinome zu ermitteln und zu prüfen. Ein einfacher Test funktioniert wie folgt:
Für jedes <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 r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>, für das <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{r}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{r}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3203a9263c30d27fcb353df376a2d1c53d6a8839.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.139ex; height:2.509ex;" alt="{\displaystyle 2^{r}-1}" loading="lazy"></span> eine <a href="Mersenne-Primzahl" class="mw-redirect" title="Mersenne-Primzahl">Mersenne-Primzahl</a> ist, ist ein Trinom des Grades <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 r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> primitiv, genau dann wenn es irreduzibel ist. Durch kürzlich von <a href="Richard_P._Brent" title="Richard P. Brent">Richard P. Brent</a> entwickelte Algorithmen ist es möglich geworden, primitive Trinome von hohem Grad zu finden, wie z. B. <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^{6972593}+X^{3037958}+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6972593</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3037958</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{6972593}+X^{3037958}+1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a602fac06d1176aec2edd332f746030c21847e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.809ex; height:2.843ex;" alt="{\displaystyle X^{6972593}+X^{3037958}+1}" loading="lazy"></span>. Damit können Pseudozufallsgeneratoren mit einer riesigen Periode von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{6972593}-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>6972593</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2^{6972593}-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d59abad13d721474dd0790420ace51eb628dc28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.151ex; height:2.843ex;" alt="{\displaystyle 2^{6972593}-1}" loading="lazy"></span>, oder ca. <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 10^{2098959}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>10</mn>
<mrow class="MJX-TeXAtom-ORD">
<mn>2098959</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 10^{2098959}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229589abe9d310b722d3ab1a43140fb900c96a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.311ex; height:2.676ex;" alt="{\displaystyle 10^{2098959}}" loading="lazy"></span> erzeugt werden.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Elwyn R. Berlekamp: <cite style="font-style:italic">Algebraic Coding Theory, Revised Edition</cite>. 2. Auflage. Aegean Park Press, 1984, ISBN 0-89412-063-8.<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:Primitives+Polynom&rft.au=Elwyn+R.+Berlekamp&rft.btitle=Algebraic+Coding+Theory%2C+Revised+Edition&rft.date=1984&rft.edition=2.&rft.genre=book&rft.isbn=0894120638&rft.pub=Aegean+Park+Press" style="display:none"> </span></li>
<li>Peterson, W.W., Weldon, E.J., "Error correcting codes", Cambridge, The MIT – Press, 1972</li>
<li>Anderson, G.C., Finnie, B. W., "Pseudo-random and random test signals", <a href="Hewlett-Packard_Journal" title="Hewlett-Packard Journal">HP Journal</a> 19, Nr. 1,2 1967</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PrimitivePolynomial.html">MathWorld entry on primitive polynomial</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Tietze/Schenk, "Halbleiter-Schaltungstechnik", 3. Auflage 1976, S.590 ff, in späteren Auflagen nicht mehr beschrieben.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://maths-people.anu.edu.au/~brent/trinom.html"><i>Search for Primitive Trinomials (mod 2).</i></a></span>
</li>
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