Polynomring – Wikipedia

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underavsnittet Polynomringar i en variabel</span> </button> <ul id="toc-Polynomringar_i_en_variabel-sublist" class="vector-toc-list"> <li id="toc-Egenskaper" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Egenskaper"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Egenskaper</span> </div> </a> <ul id="toc-Egenskaper-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomdivision" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomdivision"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Polynomdivision</span> </div> </a> <ul id="toc-Polynomdivision-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polynomringar_i_flera_variabler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polynomringar_i_flera_variabler"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Polynomringar i flera variabler</span> </div> </a> <button aria-controls="toc-Polynomringar_i_flera_variabler-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Växla underavsnittet Polynomringar i flera variabler</span> </button> <ul id="toc-Polynomringar_i_flera_variabler-sublist" class="vector-toc-list"> <li id="toc-Egenskaper_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Egenskaper_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Egenskaper</span> </div> </a> <ul id="toc-Egenskaper_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generaliseringar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generaliseringar"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generaliseringar</span> </div> </a> <button aria-controls="toc-Generaliseringar-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Växla underavsnittet Generaliseringar</span> </button> <ul id="toc-Generaliseringar-sublist" class="vector-toc-list"> <li id="toc-Generaliserade_exponenter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generaliserade_exponenter"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Generaliserade exponenter</span> </div> </a> <ul id="toc-Generaliserade_exponenter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formella_potensserier" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formella_potensserier"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Formella potensserier</span> </div> </a> <ul id="toc-Formella_potensserier-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Källor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Källor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Källor</span> </div> </a> <ul id="toc-Källor-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Innehåll" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Växla innehållsförteckningen" > <label 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class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Anell_de_polinomis" title="Anell de polinomis – katalanska" lang="ca" hreflang="ca" data-title="Anell de polinomis" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Polynomi%C3%A1ln%C3%AD_okruh" title="Polynomiální okruh – tjeckiska" lang="cs" hreflang="cs" data-title="Polynomiální okruh" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Polynomring" title="Polynomring – tyska" lang="de" hreflang="de" data-title="Polynomring" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Polynomial_ring" title="Polynomial ring – engelska" lang="en" hreflang="en" data-title="Polynomial ring" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anillo_de_polinomios" title="Anillo de polinomios – spanska" lang="es" hreflang="es" data-title="Anillo de polinomios" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D9%87_%DA%86%D9%86%D8%AF%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C" title="حلقه چندجمله‌ای – persiska" lang="fa" hreflang="fa" data-title="حلقه چندجمله‌ای" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_formel" title="Polynôme formel – franska" lang="fr" hreflang="fr" data-title="Polynôme formel" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%ED%95%AD%EC%8B%9D%ED%99%98" title="다항식환 – koreanska" lang="ko" hreflang="ko" data-title="다항식환" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Anello_de_polynomios" title="Anello de polynomios – interlingua" lang="ia" hreflang="ia" data-title="Anello de polynomios" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Anello_dei_polinomi" title="Anello dei polinomi – italienska" lang="it" hreflang="it" data-title="Anello dei polinomi" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%92_%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9E%D7%99%D7%9D" title="חוג פולינומים – hebreiska" lang="he" hreflang="he" data-title="חוג פולינומים" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Veeltermring" title="Veeltermring – nederländska" lang="nl" hreflang="nl" data-title="Veeltermring" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F%E7%92%B0" title="多項式環 – japanska" lang="ja" hreflang="ja" data-title="多項式環" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pier%C5%9Bcie%C5%84_wielomian%C3%B3w" title="Pierścień wielomianów – polska" lang="pl" hreflang="pl" data-title="Pierścień wielomianów" data-language-autonym="Polski" data-language-local-name="polska" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link 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class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">dölj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Från Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sv" dir="ltr"><p>En <b>polynomring</b> är inom <a href="/wiki/Matematik" title="Matematik">matematik</a> en <a href="/wiki/Ring_(matematik)" title="Ring (matematik)">ring</a> konstruerad från en annan ring som kan ses som mängden av alla <a href="/wiki/Polynom" title="Polynom">polynom</a> i ett fixt antal variabler med koefficienter i den ursprungliga ringen. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Polynomringar_i_en_variabel">Polynomringar i en variabel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=1" title="Redigera avsnitt: Polynomringar i en variabel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=1" title="Redigera avsnitts källkod: Polynomringar i en variabel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ett polynom i en variabel <i>x</i> med koefficienter i en ring <i>R</i> är ett uttryck på formen: </p> <dl><dd><span class="mwe-math-element"><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=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}=\sum _{k=0}^{n}a_{k}x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}=\sum _{k=0}^{n}a_{k}x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efd796d3b4fa6642f9053221f494a45be542112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:55.934ex; height:7.009ex;" alt="{\displaystyle p=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}=\sum _{k=0}^{n}a_{k}x^{k}}"></span></dd></dl> <p>där <span class="mwe-math-element"><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_{n},a_{n-1},...,a_{1},a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n},a_{n-1},...,a_{1},a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e594ec42dc69424cc79c6ffc95deceacb94a43a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.803ex; height:2.009ex;" alt="{\displaystyle a_{n},a_{n-1},...,a_{1},a_{0}}"></span> är element i <i>R</i>. Med <b>graden av <i>p</i></b> avses det största <i>k</i> sådant att <span class="mwe-math-element"><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^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525e1133440e1565055dec6243aaf0f27d4d4e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}"></span> har en nollskild koefficient. </p><p><b>Polynomringen över <i>R</i></b>, betecknad <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> mängden av alla polynom med koefficienter i <i>R</i>. <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> är då en ring med operatorerna addition och multiplikation definierade enligt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{k=0}^{n}a_{k}x^{k}\right)+\left(\sum _{k=0}^{n}b_{k}x^{k}\right)=\sum _{k=0}^{n}(a_{k}+b_{k})x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{k=0}^{n}a_{k}x^{k}\right)+\left(\sum _{k=0}^{n}b_{k}x^{k}\right)=\sum _{k=0}^{n}(a_{k}+b_{k})x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d39faa763fac8287aac49ace104fa03eb17904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.855ex; height:7.509ex;" alt="{\displaystyle \left(\sum _{k=0}^{n}a_{k}x^{k}\right)+\left(\sum _{k=0}^{n}b_{k}x^{k}\right)=\sum _{k=0}^{n}(a_{k}+b_{k})x^{k}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{i=0}^{n}a_{i}x^{i}\right)\left(\sum _{j=0}^{m}b_{j}x^{j}\right)=\sum _{k=0}^{m+n}\left(\sum _{i+j=k}a_{i}b_{j}\right)x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{i=0}^{n}a_{i}x^{i}\right)\left(\sum _{j=0}^{m}b_{j}x^{j}\right)=\sum _{k=0}^{m+n}\left(\sum _{i+j=k}a_{i}b_{j}\right)x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0ba893ab338751ad7fe3cfbab8a46f24af0ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.202ex; height:7.676ex;" alt="{\displaystyle \left(\sum _{i=0}^{n}a_{i}x^{i}\right)\left(\sum _{j=0}^{m}b_{j}x^{j}\right)=\sum _{k=0}^{m+n}\left(\sum _{i+j=k}a_{i}b_{j}\right)x^{k}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Egenskaper">Egenskaper</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=2" title="Redigera avsnitt: Egenskaper" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=2" title="Redigera avsnitts källkod: Egenskaper"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Om <i>R</i> är en <a href="/wiki/Kommutativ_ring" title="Kommutativ ring">kommutativ ring</a> är <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> en kommutativ ring.</li> <li>Om <i>R</i> är ett <a href="/wiki/Integritetsomr%C3%A5de" title="Integritetsområde">integritetsområde</a> är <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> ett integritetsområde.</li> <li>Om <i>K</i> är en <a href="/wiki/Kropp_(algebra)" title="Kropp (algebra)">kropp</a> är <span class="mwe-math-element"><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]}"> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9e6c2ac2830d6a9abe078b47450777c41d69a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle K[x]}"></span> en <a href="/wiki/Principalidealdom%C3%A4n" title="Principalidealdomän">principalidealdomän</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Polynomdivision">Polynomdivision</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=3" title="Redigera avsnitt: Polynomdivision" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=3" title="Redigera avsnitts källkod: Polynomdivision"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Om <i>d</i> är ett element i <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> vars ledande koefficient är en <a href="/w/index.php?title=Enhet_(ringteori)&action=edit&redlink=1" class="new" title="Enhet (ringteori) [inte skriven än]">enhet</a> i <i>R</i> (ett inverterbart element) så finns för alla <i>p</i> i <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> unika element <i>k</i> och <i>r</i> i <span class="mwe-math-element"><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[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce54622cb380383ab3a42441b056626ea0d2440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.387ex; height:2.843ex;" alt="{\displaystyle R[x]}"></span> sådana att <i>k</i>:s grad är strikt mindre än <i>r</i>:s grad och </p> <dl><dd><span class="mwe-math-element"><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=kd+r.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>k</mi> <mi>d</mi> <mo>+</mo> <mi>r</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=kd+r.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cd3e58f54d3fe0b20ae8cea96bb1c8704fa3de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.707ex; height:2.509ex;" alt="{\displaystyle p=kd+r.\,}"></span></dd></dl> <p>Speciellt, om <i>K</i> är en kropp gäller ovan för alla element <i>d</i> i <span class="mwe-math-element"><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]}"> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9e6c2ac2830d6a9abe078b47450777c41d69a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle K[x]}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Polynomringar_i_flera_variabler">Polynomringar i flera variabler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=4" title="Redigera avsnitt: Polynomringar i flera variabler" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=4" title="Redigera avsnitts källkod: Polynomringar i flera variabler"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ett polynom i flera variabler <span class="mwe-math-element"><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},...,x_{n}}"> <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> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},...,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f979c14353ba9d99b39d68265ad6db58c5faaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.009ex;" alt="{\displaystyle x_{1},...,x_{n}}"></span> med koefficienter i en ring <i>R</i> definieras analogt med polynom i en variabel, men notationen är omständligare. Vanligtvis definieras ett multiindex <span class="mwe-math-element"><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 =(\alpha _{1},...,\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =(\alpha _{1},...,\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91dccfbcc710f862385e9574388c7a61b4fa1cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.813ex; height:2.843ex;" alt="{\displaystyle \alpha =(\alpha _{1},...,\alpha _{n})}"></span> som är en n-tippel av heltal <span class="mwe-math-element"><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 _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1fb627423abe4988b7ed88d4920bf1ec074790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}"></span> och man skriver: </p> <dl><dd><span class="mwe-math-element"><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^{\alpha }=\prod _{k=1}^{n}x_{k}^{\alpha _{k}}=x_{1}^{\alpha _{1}}\ldots x_{n}^{\alpha _{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\alpha }=\prod _{k=1}^{n}x_{k}^{\alpha _{k}}=x_{1}^{\alpha _{1}}\ldots x_{n}^{\alpha _{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab84204e9218bf0bdf9a71441b90a46eba4459f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.162ex; height:6.843ex;" alt="{\displaystyle x^{\alpha }=\prod _{k=1}^{n}x_{k}^{\alpha _{k}}=x_{1}^{\alpha _{1}}\ldots x_{n}^{\alpha _{n}}}"></span></dd></dl> <p>och produkten <span class="mwe-math-element"><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^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ecb06c51aa5eb67cb4363751c8774977ba3cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.614ex; height:2.343ex;" alt="{\displaystyle x^{\alpha }}"></span> kallas för ett <a href="/wiki/Monom" title="Monom">monom</a> av <b>multigrad</b> <span class="mwe-math-element"><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</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><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 }"></span>. Ett <b>polynom över <i>R</i></b> definieras då som en <a href="/wiki/Linj%C3%A4rkombination" title="Linjärkombination">linjärkombination</a> av monom med koefficienter i <i>R</i>: </p> <dl><dd><span class="mwe-math-element"><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=\sum _{\alpha }a_{\alpha }x^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\sum _{\alpha }a_{\alpha }x^{\alpha }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4549a2af244dcbcb21776317806a459bb841099e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:13.874ex; height:5.509ex;" alt="{\displaystyle p=\sum _{\alpha }a_{\alpha }x^{\alpha }.}"></span></dd></dl> <p>Med <b>graden</b> av ett monom <span class="mwe-math-element"><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^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ecb06c51aa5eb67cb4363751c8774977ba3cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.614ex; height:2.343ex;" alt="{\displaystyle x^{\alpha }}"></span> avses: </p> <dl><dd><span class="mwe-math-element"><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 |=\sum _{k=1}^{n}\alpha _{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha |=\sum _{k=1}^{n}\alpha _{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43eca4072c4d2bb8bd793c5e2a40b9061c8e80b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.845ex; height:6.843ex;" alt="{\displaystyle |\alpha |=\sum _{k=1}^{n}\alpha _{k}.}"></span></dd></dl> <p>En <b>polynomring i <i>n</i> variabler över <i>R</i></b>, <span class="mwe-math-element"><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[x_{1},...,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</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> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x_{1},...,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf01aa2b22aec9bbbbf89b5d87aa477d6e57661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.513ex; height:2.843ex;" alt="{\displaystyle R[x_{1},...,x_{n}}"></span> är alla polynom med <i>n</i> variabler, dessa kan konstrueras genom att skapa polynomringar av polynomringar, exempelvis är <span class="mwe-math-element"><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[x_{1},x_{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</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 stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[x_{1},x_{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988814d37b70fd9a5de5a6b16c12ac44af2bac91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.859ex; height:2.843ex;" alt="{\displaystyle R[x_{1},x_{2}]}"></span> <a href="/w/index.php?title=Ringisomorfi&action=edit&redlink=1" class="new" title="Ringisomorfi [inte skriven än]">isomorf</a> med <span class="mwe-math-element"><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[R[x]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[R[x]]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc88647dca40c977f789a0427670266a597dcbe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.445ex; height:2.843ex;" alt="{\displaystyle R[R[x]]}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Egenskaper_2">Egenskaper</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=5" title="Redigera avsnitt: Egenskaper" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=5" title="Redigera avsnitts källkod: Egenskaper"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Låt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R[x_{1},...,x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>R</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> <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 S=R[x_{1},...,x_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/853dd3666d41d58711b26548e0b3800874a807b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.757ex; height:2.843ex;" alt="{\displaystyle S=R[x_{1},...,x_{n}]}"></span> där <i>R</i> är en ring. Då gäller: </p> <ul><li>Om <i>R</i> är kommutativ är <i>S</i> kommutativ.</li> <li>Om <i>R</i> är ett integritetsområde är <i>S</i> ett integritetsområde.</li> <li>Om <i>R</i> är en kropp är alla <a href="/wiki/Ideal_(ringteori)" title="Ideal (ringteori)">ideal</a> i <i>S</i> <a href="/w/index.php?title=%C3%84ndligt_genererade&action=edit&redlink=1" class="new" title="Ändligt genererade [inte skriven än]">ändligt genererade</a> (<a href="/wiki/Hilberts_bassats" title="Hilberts bassats">Hilberts bassats</a>).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Generaliseringar">Generaliseringar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=6" title="Redigera avsnitt: Generaliseringar" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=6" title="Redigera avsnitts källkod: Generaliseringar"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polynomringar kan generaliseras på flera olika sätt. </p> <div class="mw-heading mw-heading3"><h3 id="Generaliserade_exponenter">Generaliserade exponenter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=7" title="Redigera avsnitt: Generaliserade exponenter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=7" title="Redigera avsnitts källkod: Generaliserade exponenter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I en polynomring är exponenterna på variablerna heltal, men den avgörande egenskapen för att strukturen ska bli en ring är sambandet </p> <dl><dd><span class="mwe-math-element"><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^{m}x^{n}=x^{m+n}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{m}x^{n}=x^{m+n}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eccac88e76208da86d671c8110c45b75b49ee0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.955ex; height:2.509ex;" alt="{\displaystyle x^{m}x^{n}=x^{m+n}\,.}"></span></dd></dl> <p>Dvs, att man kan lägga ihop exponenter, en operation som är associativ. En struktur med en <a href="/wiki/Bin%C3%A4r_operator" title="Binär operator">binär operator</a> som är associativ kallas för en <a href="/wiki/Monoid" title="Monoid">monoid</a>. Mängden av funktioner med <a href="/wiki/Nollskild" class="mw-redirect" title="Nollskild">nollskilda</a> värden för endast ändligt många element från en monoid <i>M</i> till en ring <i>R</i> bildar en så kallad <a href="/w/index.php?title=Monoidring&action=edit&redlink=1" class="new" title="Monoidring [inte skriven än]">monoidring</a>, <span class="mwe-math-element"><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[N]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[N]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32f39f8b5f00df6a8ca8d79288f62bb82f04054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.121ex; height:2.843ex;" alt="{\displaystyle R[N]}"></span>. En polynomring i <i>n</i> variabler över <i>R</i> är en monoidring <span class="mwe-math-element"><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[\mathbb {N} ^{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\mathbb {N} ^{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16006a990db080df6a8b57a1275dd05cab494ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.954ex; height:2.843ex;" alt="{\displaystyle R[\mathbb {N} ^{n}]}"></span>, där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de4673cd92dd1c8e3a19aeda306b77ad113ebd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {N} ^{n}}"></span> är monoiden <i>n</i>-tipplar av <a href="/wiki/Naturliga_tal" title="Naturliga tal">naturliga tal</a> med addition som binär operator. Man kan utgå från definitionen av en monoidring och konstruera begreppet polynomring som ett specialfall. Andra val av monoider än <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de4673cd92dd1c8e3a19aeda306b77ad113ebd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {N} ^{n}}"></span> ger andra typer av monoidringar. </p> <div class="mw-heading mw-heading3"><h3 id="Formella_potensserier">Formella potensserier</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=8" title="Redigera avsnitt: Formella potensserier" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=8" title="Redigera avsnitts källkod: Formella potensserier"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Istället för polynom kan man använda <a href="/w/index.php?title=Formell_potensserie&action=edit&redlink=1" class="new" title="Formell potensserie [inte skriven än]">formella potensserier</a> som sina ringelement, då man kan ha oändligt många nollskilda koefficienter. Addition sker komponentvis och multiplikation genom <a href="/w/index.php?title=Cauchyprodukt&action=edit&redlink=1" class="new" title="Cauchyprodukt [inte skriven än]">Cauchyprodukten</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Källor"><span id="K.C3.A4llor"></span>Källor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polynomring&veaction=edit&section=9" title="Redigera avsnitt: Källor" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polynomring&action=edit&section=9" title="Redigera avsnitts källkod: Källor"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal" class="book" id="CITEREFGrillet2007">Grillet, Pierre Antoine (2007). <i><span>Abstract Algebra</span></i>. Springer Verlag. <a href="/wiki/Special:Bokk%C3%A4llor/978-0-387-71567-4" title="Special:Bokkällor/978-0-387-71567-4">ISBN 978-0-387-71567-4</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra&rft.aulast=Grillet&rft.aufirst=Pierre+Antoine&rft.au=Grillet%2C+Pierre+Antoine&rft.date=2007&rft.pub=Springer+Verlag&rft.isbn=978-0-387-71567-4&rfr_id=info:sid/en.wikipedia.org:Polynomring"><span style="display: none;"> </span></span></li> <li><cite style="font-style:normal" class="book" id="CITEREFLang2002">Lang, Serge (2002). <i><span>Algebra</span></i>. Springer Verlag. <a href="/wiki/Special:Bokk%C3%A4llor/0-387-95385-X" title="Special:Bokkällor/0-387-95385-X">ISBN 0-387-95385-X</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.aulast=Lang&rft.aufirst=Serge&rft.au=Lang%2C+Serge&rft.date=2002&rft.pub=Springer+Verlag&rft.isbn=0-387-95385-X&rfr_id=info:sid/en.wikipedia.org:Polynomring"><span style="display: none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Hämtad från ”<a dir="ltr" href="https://sv.wikipedia.org/w/index.php?title=Polynomring&oldid=50600287">https://sv.wikipedia.org/w/index.php?title=Polynomring&oldid=50600287</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorier" title="Wikipedia:Kategorier">Kategorier</a>: <ul><li><a href="/wiki/Kategori:Ringteori" title="Kategori:Ringteori">Ringteori</a></li><li><a href="/wiki/Kategori:Invariantteori" title="Kategori:Invariantteori">Invariantteori</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Sidan redigerades senast den 16 juni 2022 kl. 18.27.</li> <li id="footer-info-copyright">Wikipedias text är tillgänglig under licensen <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.sv">Creative Commons Erkännande-dela-lika 4.0 Unported</a>. 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Polynomring – Wikipedia