Polynomial ring - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Polynomial ring - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"e42b40d9-c5e7-4f80-9c68-8a340ce4a014","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Polynomial_ring","wgTitle":"Polynomial ring","wgCurRevisionId":1254302004,"wgRevisionId":1254302004,"wgArticleId":373065,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Articles needing cleanup from June 2023","All pages needing cleanup","Cleanup tagged articles with a reason field from June 2023","Wikipedia pages needing cleanup from June 2023","Articles needing additional references from January 2021","All articles needing additional references","Articles to be expanded from April 2022","All articles to be expanded","Articles with empty sections from April 2022", "All articles with empty sections","Articles to be expanded from June 2020","Harv and Sfn no-target errors","Commutative algebra","Invariant theory","Ring theory","Polynomials","Free algebraic structures"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Polynomial_ring","wgRelevantArticleId":373065,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain": false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1455652","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready", "ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Polynomial ring - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Polynomial_ring"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Polynomial_ring&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Polynomial_ring"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Polynomial_ring rootpage-Polynomial_ring skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Polynomial+ring" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Polynomial+ring" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Polynomial+ring" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Polynomial+ring" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition_(univariate_case)" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_(univariate_case)"> <div class="vector-toc-text"> 1 Definition (univariate case) </div> </a> <button aria-controls="toc-Definition_(univariate_case)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition (univariate case) subsection </button> <ul id="toc-Definition_(univariate_case)-sublist" class="vector-toc-list"> <li id="toc-Terminology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Terminology"> <div class="vector-toc-text"> 1.1 Terminology </div> </a> <ul id="toc-Terminology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_evaluation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_evaluation"> <div class="vector-toc-text"> 1.2 Polynomial evaluation </div> </a> <ul id="toc-Polynomial_evaluation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Univariate_polynomials_over_a_field" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Univariate_polynomials_over_a_field"> <div class="vector-toc-text"> 2 Univariate polynomials over a field </div> </a> <button aria-controls="toc-Univariate_polynomials_over_a_field-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Univariate polynomials over a field subsection </button> <ul id="toc-Univariate_polynomials_over_a_field-sublist" class="vector-toc-list"> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> 2.1 Derivation </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> <li id="toc-Square-free_factorization" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Square-free_factorization"> <div class="vector-toc-text"> 2.1.1 Square-free factorization </div> </a> <ul id="toc-Square-free_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lagrange_interpolation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lagrange_interpolation"> <div class="vector-toc-text"> 2.1.2 Lagrange interpolation </div> </a> <ul id="toc-Lagrange_interpolation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_decomposition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Polynomial_decomposition"> <div class="vector-toc-text"> 2.1.3 Polynomial decomposition </div> </a> <ul id="toc-Polynomial_decomposition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Factorization"> <div class="vector-toc-text"> 2.2 Factorization </div> </a> <ul id="toc-Factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimal_polynomial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimal_polynomial"> <div class="vector-toc-text"> 2.3 Minimal polynomial </div> </a> <ul id="toc-Minimal_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quotient_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quotient_ring"> <div class="vector-toc-text"> 2.4 Quotient ring </div> </a> <ul id="toc-Quotient_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modules"> <div class="vector-toc-text"> 2.5 Modules </div> </a> <ul id="toc-Modules-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition_(multivariate_case)" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_(multivariate_case)"> <div class="vector-toc-text"> 3 Definition (multivariate case) </div> </a> <button aria-controls="toc-Definition_(multivariate_case)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition (multivariate case) subsection </button> <ul id="toc-Definition_(multivariate_case)-sublist" class="vector-toc-list"> <li id="toc-Operations_in_K[X1,_...,_Xn]" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operations_in_K[X1,_...,_Xn]"> <div class="vector-toc-text"> 3.1 Operations in K[X1, ..., Xn] </div> </a> <ul id="toc-Operations_in_K[X1,_...,_Xn]-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_expression"> <div class="vector-toc-text"> 3.2 Polynomial expression </div> </a> <ul id="toc-Polynomial_expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Categorical_characterization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Categorical_characterization"> <div class="vector-toc-text"> 3.3 Categorical characterization </div> </a> <ul id="toc-Categorical_characterization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graded_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Graded_structure"> <div class="vector-toc-text"> 4 Graded structure </div> </a> <ul id="toc-Graded_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Univariate_over_a_ring_vs._multivariate" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Univariate_over_a_ring_vs._multivariate"> <div class="vector-toc-text"> 5 Univariate over a ring vs. multivariate </div> </a> <button aria-controls="toc-Univariate_over_a_ring_vs._multivariate-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Univariate over a ring vs. multivariate subsection </button> <ul id="toc-Univariate_over_a_ring_vs._multivariate-sublist" class="vector-toc-list"> <li id="toc-Properties_that_pass_from_R_to_R[X]" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_that_pass_from_R_to_R[X]"> <div class="vector-toc-text"> 5.1 Properties that pass from R to R[X] </div> </a> <ul id="toc-Properties_that_pass_from_R_to_R[X]-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Several_indeterminates_over_a_field" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Several_indeterminates_over_a_field"> <div class="vector-toc-text"> 6 Several indeterminates over a field </div> </a> <button aria-controls="toc-Several_indeterminates_over_a_field-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Several indeterminates over a field subsection </button> <ul id="toc-Several_indeterminates_over_a_field-sublist" class="vector-toc-list"> <li id="toc-Hilbert's_Nullstellensatz" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert's_Nullstellensatz"> <div class="vector-toc-text"> 6.1 Hilbert's Nullstellensatz </div> </a> <ul id="toc-Hilbert's_Nullstellensatz-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bézout's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bézout's_theorem"> <div class="vector-toc-text"> 6.2 Bézout's theorem </div> </a> <ul id="toc-Bézout's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jacobian_conjecture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jacobian_conjecture"> <div class="vector-toc-text"> 6.3 Jacobian conjecture </div> </a> <ul id="toc-Jacobian_conjecture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 7 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Infinitely_many_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinitely_many_variables"> <div class="vector-toc-text"> 7.1 Infinitely many variables </div> </a> <ul id="toc-Infinitely_many_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_exponents" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalized_exponents"> <div class="vector-toc-text"> 7.2 Generalized exponents </div> </a> <ul id="toc-Generalized_exponents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_series"> <div class="vector-toc-text"> 7.3 Power series </div> </a> <ul id="toc-Power_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noncommutative_polynomial_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noncommutative_polynomial_rings"> <div class="vector-toc-text"> 7.4 Noncommutative polynomial rings </div> </a> <ul id="toc-Noncommutative_polynomial_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_and_skew-polynomial_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_and_skew-polynomial_rings"> <div class="vector-toc-text"> 7.5 Differential and skew-polynomial rings </div> </a> <ul id="toc-Differential_and_skew-polynomial_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_rigs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_rigs"> <div class="vector-toc-text"> 7.6 Polynomial rigs </div> </a> <ul id="toc-Polynomial_rigs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-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="Contents" 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="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Polynomial ring</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 22 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-22" aria-hidden="true" > 22 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D8%A9_%D9%85%D8%AA%D8%B9%D8%AF%D8%AF%D8%A7%D8%AA_%D8%A7%D9%84%D8%AD%D8%AF%D9%88%D8%AF" title="حلقة متعددات الحدود – Arabic" lang="ar" hreflang="ar" data-title="حلقة متعددات الحدود" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Anell_de_polinomis" title="Anell de polinomis – Catalan" lang="ca" hreflang="ca" data-title="Anell de polinomis" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</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 – Czech" lang="cs" hreflang="cs" data-title="Polynomiální okruh" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Polynomring" title="Polynomring – German" lang="de" hreflang="de" data-title="Polynomring" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</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 – Spanish" lang="es" hreflang="es" data-title="Anillo de polinomios" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</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 – Persian" lang="fa" hreflang="fa" data-title="حلقه چندجمله‌ای" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</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 – French" lang="fr" hreflang="fr" data-title="Polynôme formel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</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 – Korean" lang="ko" hreflang="ko" data-title="다항식환" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</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">Interlingua</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 – Italian" lang="it" hreflang="it" data-title="Anello dei polinomi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="חוג פולינומים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Veeltermring" title="Veeltermring – Dutch" lang="nl" hreflang="nl" data-title="Veeltermring" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="多項式環" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</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 – Polish" lang="pl" hreflang="pl" data-title="Pierścień wielomianów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Anel_de_polin%C3%B4mios" title="Anel de polinômios – Portuguese" lang="pt" hreflang="pt" data-title="Anel de polinômios" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD%D0%BE%D0%B2" title="Кольцо многочленов – Russian" lang="ru" hreflang="ru" data-title="Кольцо многочленов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Polynomirengas" title="Polynomirengas – Finnish" lang="fi" hreflang="fi" data-title="Polynomirengas" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Polynomring" title="Polynomring – Swedish" lang="sv" hreflang="sv" data-title="Polynomring" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%96%D0%BB%D1%8C%D1%86%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD%D1%96%D0%B2" title="Кільце многочленів – Ukrainian" lang="uk" hreflang="uk" data-title="Кільце многочленів" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%C3%A0nh_%C4%91a_th%E1%BB%A9c" title="Vành đa thức – Vietnamese" lang="vi" hreflang="vi" data-title="Vành đa thức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A0%B4%E4%B8%8A%E5%A4%9A%E9%A0%85%E5%BC%8F" title="場上多項式 – Cantonese" lang="yue" hreflang="yue" data-title="場上多項式" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%9A%E9%A1%B9%E5%BC%8F%E7%8E%AF" title="多项式环 – Chinese" lang="zh" hreflang="zh" data-title="多项式环" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1455652#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Polynomial_ring" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Polynomial_ring" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Polynomial_ring">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Polynomial_ring&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Polynomial_ring&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Polynomial_ring">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Polynomial_ring&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Polynomial_ring&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Polynomial_ring" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Polynomial_ring" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Polynomial_ring&oldid=1254302004" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Polynomial_ring&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Polynomial_ring&id=1254302004&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPolynomial_ring">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPolynomial_ring">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Polynomial_ring&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Polynomial_ring&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Abstract_Algebra/Polynomial_Rings" hreflang="en">Wikibooks</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1455652" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div 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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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">From Wikipedia, the free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebraic structure</div><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a> <dl><dd>• <a href="/wiki/Subring" title="Subring">Subrings</a></dd> <dd>• <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd> <dd>• <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a> <dl><dd>• <a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd> <dd>• <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd> <dd>• <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></dd> <dd>• <a href="/wiki/Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd> <dd>• <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl> <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a> <dl><dd>• <a href="/wiki/Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd> <dd>• <a href="/wiki/Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd> <dd>• <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl> <a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> <dl><dd>• <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></dd> <dd>• <a href="/wiki/Associative_algebra" title="Associative algebra">Associative algebra</a></dd> <dd>• <a href="/wiki/Graded_ring" title="Graded ring">Graded ring</a></dd> <dd>• <a href="/wiki/Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd> <dd>• <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a> <dl><dd>• <a href="/wiki/Integer" title="Integer">Initial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ab495cb8cfbac68a9322af662c3d6c7dbe494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"></dd></dl></dd></dl> Related structures <dl><dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a> <dl><dd>• <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></dd> <dd>• <a href="/wiki/Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd> <dd>• <a href="/wiki/Semiring" title="Semiring">Semiring</a> <dl><dd>• <a href="/wiki/Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a> <dl><dd>• <a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a> <dl><dd>• <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd> <dd>• <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a></dd> <dd>• <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd> <dd>• <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd> <dd>• <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd> <dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a class="mw-selflink selflink">Polynomial ring</a></dd> <dd>• <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl> <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> <dl><dd>• <a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo n</a></dd> <dd>• <a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></dd> <dd>• <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer p-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14af623e08c241266c125ad927dd35086ec8ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}"></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a> <dl><dd>• <a href="/wiki/Division_ring" title="Division ring">Division ring</a></dd> <dd>• <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd> <dd>• <a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a></dd> <dd>• <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl> <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a> <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a> <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Ring_theory_sidebar" title="Template:Ring theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ring_theory_sidebar" title="Template talk:Ring theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ring_theory_sidebar" title="Special:EditPage/Template:Ring theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, especially in the field of <a href="/wiki/Algebra" title="Algebra">algebra</a>, a polynomial ring or polynomial algebra is a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> formed from the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> in one or more <a href="/wiki/Indeterminate_(variable)" title="Indeterminate (variable)">indeterminates</a> (traditionally also called <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>) with <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in another <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, often a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>.<a href="#cite_note-1">[1]</a> Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the <a href="/wiki/Integer#Algebraic_properties" title="Integer">integers</a>.<a href="#cite_note-:0-2">[2]</a> Polynomial rings occur and are often fundamental in many parts of mathematics such as <a href="/wiki/Number_theory" title="Number theory">number theory</a>, <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. In <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>, many classes of rings, such as <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a>, <a href="/wiki/Regular_ring" class="mw-redirect" title="Regular ring">regular rings</a>, <a href="/wiki/Group_ring" title="Group ring">group rings</a>, <a href="/wiki/Formal_power_series" title="Formal power series">rings of formal power series</a>, <a href="/wiki/Ore_polynomial" class="mw-redirect" title="Ore polynomial">Ore polynomials</a>, <a href="/wiki/Graded_ring" title="Graded ring">graded rings</a>, have been introduced for generalizing some properties of polynomial rings.<a href="#cite_note-3">[3]</a> A closely related notion is that of the <a href="/wiki/Ring_of_polynomial_functions" title="Ring of polynomial functions">ring of polynomial functions</a> on a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, and, more generally, <a href="/wiki/Ring_of_regular_functions" class="mw-redirect" title="Ring of regular functions">ring of regular functions</a> on an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>.<a href="#cite_note-:0-2">[2]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_(univariate_case)">Definition (univariate case)</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=1" title="Edit section: Definition (univariate case)">edit</a>]</div> Let K be a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or (more generally) a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>. The polynomial ring in X over K, which is denoted K[X], can be defined in several equivalent ways. One of them is to define K[X] as the set of expressions, called polynomials in X, of the form<a href="#cite_note-4">[4]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>p</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a98a1580574bb492732b0c7b39ea8654b64c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:50.865ex; height:3.009ex;" alt="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}"></dd></dl> where p0, p1, …, pm, the coefficients of p, are elements of K, pm ≠ 0 if m > 0, and X, X2, …, are symbols, which are considered as "powers" of X, and follow the usual rules of <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>: X0 = 1, X1 = X, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{k}\,X^{l}=X^{k+l}}"> <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> <mspace width="thinmathspace" /> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>l</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{k}\,X^{l}=X^{k+l}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c69dd2ebb14e1afddc096c63f7993c7e3583b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.145ex; height:2.676ex;" alt="{\displaystyle X^{k}\,X^{l}=X^{k+l}}"> for any <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integers</a> k and l. The symbol X is called an indeterminate<a href="#cite_note-5">[5]</a> or variable.<a href="#cite_note-6">[6]</a> (The term of "variable" comes from the terminology of <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a>. However, here, X has no value (other than itself), and cannot vary, being a constant in the polynomial ring.) Two polynomials are equal when the corresponding coefficients of each Xk are equal. One can think of the ring K[X] as arising from K by adding one new element X that is external to K, commutes with all elements of K, and has no other specific properties. This can be used for an equivalent definition of polynomial rings. The polynomial ring in X over K is equipped with an addition, a multiplication and a <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> that make it a <a href="/wiki/Commutative_algebra_(structure)" class="mw-redirect" title="Commutative algebra (structure)">commutative algebra</a>. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>p</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711b2876358cbe83aac16bf947d961d3e818230f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:37.307ex; height:3.009ex;" alt="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m},}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>q</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>q</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f0ed4fbf183ca81f8f41d979ce6898f60e594c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.675ex; height:3.009ex;" alt="{\displaystyle q=q_{0}+q_{1}X+q_{2}X^{2}+\cdots +q_{n}X^{n},}"></dd></dl> then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>r</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>r</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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212799bcd3c47b0685ce5c9509d302d33838d3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:39.562ex; height:3.009ex;" alt="{\displaystyle p+q=r_{0}+r_{1}X+r_{2}X^{2}+\cdots +r_{k}X^{k},}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>q</mi> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>s</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43415c6a0f0b41cd58ab70d94ba9d4bba6ed4dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:36.156ex; height:3.009ex;" alt="{\displaystyle pq=s_{0}+s_{1}X+s_{2}X^{2}+\cdots +s_{l}X^{l},}"></dd></dl> where k = max(m, n), l = m + n, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}=p_{i}+q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}=p_{i}+q_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/820e0847b3d24cf28bd9cd6fd7432dbe1a71c009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.593ex; height:2.343ex;" alt="{\displaystyle r_{i}=p_{i}+q_{i}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad837fcb4f1dead70c7ab66c140a86cc6b1fc9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.162ex; height:2.343ex;" alt="{\displaystyle s_{i}=p_{0}q_{i}+p_{1}q_{i-1}+\cdots +p_{i}q_{0}.}"></dd></dl> In these formulas, the polynomials p and q are extended by adding "dummy terms" with zero coefficients, so that all pi and qi that appear in the formulas are defined. Specifically, if m < n, then pi = 0 for m < i ≤ n. The scalar multiplication is the special case of the multiplication where p = p0 is reduced to its constant term (the term that is independent of X); that is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}\left(q_{0}+q_{1}X+\dots +q_{n}X^{n}\right)=p_{0}q_{0}+\left(p_{0}q_{1}\right)X+\cdots +\left(p_{0}q_{n}\right)X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>q</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> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>X</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}\left(q_{0}+q_{1}X+\dots +q_{n}X^{n}\right)=p_{0}q_{0}+\left(p_{0}q_{1}\right)X+\cdots +\left(p_{0}q_{n}\right)X^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f24f40ae70799b47b9bc9d23a862398813af44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:64.427ex; height:2.843ex;" alt="{\displaystyle p_{0}\left(q_{0}+q_{1}X+\dots +q_{n}X^{n}\right)=p_{0}q_{0}+\left(p_{0}q_{1}\right)X+\cdots +\left(p_{0}q_{n}\right)X^{n}}"></dd></dl> It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over K. Therefore, polynomial rings are also called polynomial algebras. Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite <a href="/wiki/Sequence" title="Sequence">sequence</a> (p0, p1, p2, …) of elements of K, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some m so that pn = 0 for n > m. In this case, p0 and X are considered as alternate notations for the sequences (p0, 0, 0, …) and (0, 1, 0, 0, …), respectively. A straightforward use of the operation rules shows that the expression <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>p</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b97e40e088d4e02ffe6dafdaa9d1bbc28c067fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:32.393ex; height:3.009ex;" alt="{\displaystyle p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m}X^{m}}"></dd></dl> is then an alternate notation for the sequence <dl><dd>(p0, p1, p2, …, pm, 0, 0, …).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Terminology">Terminology</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=2" title="Edit section: Terminology">edit</a>]</div> Let <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <msub> <mi>p</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> <mo>⋯</mo> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a98a1580574bb492732b0c7b39ea8654b64c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:50.865ex; height:3.009ex;" alt="{\displaystyle p=p_{0}+p_{1}X+p_{2}X^{2}+\cdots +p_{m-1}X^{m-1}+p_{m}X^{m},}"></dd></dl> be a nonzero polynomial with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00dacba9af9ae12355de9d20799276cdc5e9a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.195ex; height:2.676ex;" alt="{\displaystyle p_{m}\neq 0}"> The constant term of p is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6a990c974dd2af4bc7a4f28165f8e151895eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle p_{0}.}"> It is zero in the case of the zero polynomial. The degree of p, written deg(p) is <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad66d19bb37bc69223cb004be2ea5dd95f9564c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.687ex; height:2.009ex;" alt="{\displaystyle m,}"> the largest k such that the coefficient of Xk is not zero.<a href="#cite_note-7">[7]</a> The leading coefficient of p is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{m}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38525d102a41271ff9ea650117ab3185bbc99d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.581ex; height:2.009ex;" alt="{\displaystyle p_{m}.}"><a href="#cite_note-8">[8]</a> In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined,<a href="#cite_note-9">[9]</a> defined to be −1,<a href="#cite_note-10">[10]</a> or defined to be a −∞.<a href="#cite_note-11">[11]</a> A constant polynomial is either the zero polynomial, or a polynomial of degree zero. A nonzero polynomial is <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic</a> if its leading coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}"> Given two polynomials p and q, if the degree of the zero polynomial is defined to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d5ae78431259d0588140f827d0a1fcdfa4ec7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.779ex; height:2.343ex;" alt="{\displaystyle -\infty ,}"> one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(p+q)\leq \max(\deg(p),\deg(q)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(p+q)\leq \max(\deg(p),\deg(q)),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d799c7bc1b1d5b6ab82844fd7ff229e92fcafa01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.123ex; height:2.843ex;" alt="{\displaystyle \deg(p+q)\leq \max(\deg(p),\deg(q)),}"></dd></dl> and, over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, or more generally an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>,<a href="#cite_note-12">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(pq)=\deg(p)+\deg(q).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(pq)=\deg(p)+\deg(q).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda60db0a9829e4489a7f2951746dd377e2942d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.954ex; height:2.843ex;" alt="{\displaystyle \deg(pq)=\deg(p)+\deg(q).}"></dd></dl> It follows immediately that, if K is an integral domain, then so is K[X].<a href="#cite_note-13">[13]</a> It follows also that, if K is an integral domain, a polynomial is a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> (that is, it has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>) if and only if it is constant and is a unit in K. Two polynomials are <a href="/wiki/Associated_element" class="mw-redirect" title="Associated element">associated</a> if either one is the product of the other by a unit. Over a field, every nonzero polynomial is associated to a unique monic polynomial. Given two polynomials, p and q, one says that p divides q, p is a divisor of q, or q is a multiple of p, if there is a polynomial r such that q = pr. A polynomial is <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree. <div class="mw-heading mw-heading3"><h3 id="Polynomial_evaluation">Polynomial evaluation</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=3" title="Edit section: Polynomial evaluation">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Polynomial_evaluation" title="Polynomial evaluation">Polynomial evaluation</a></div> Let K be a field or, more generally, a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, and R a ring containing K. For any polynomial P in K[X] and any element a in R, the substitution of X with a in P defines an element of R, which is <a href="/wiki/Polynomial_notation" class="mw-redirect" title="Polynomial notation">denoted</a> P(a). This element is obtained by carrying on in R after the substitution the operations indicated by the expression of the polynomial. This computation is called the evaluation of P at a. For example, if we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=X^{2}-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=X^{2}-1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d72ad31270fc7a6ba44669739a801a13b4edaff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.545ex; height:3.009ex;" alt="{\displaystyle P=X^{2}-1,}"></dd></dl> we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P(3)&=3^{2}-1=8,\\P(X^{2}+1)&=\left(X^{2}+1\right)^{2}-1=X^{4}+2X^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(3)&=3^{2}-1=8,\\P(X^{2}+1)&=\left(X^{2}+1\right)^{2}-1=X^{4}+2X^{2}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ded8d535505a93146c397bad1cfdee8888c50b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.751ex; margin-bottom: -0.253ex; width:41.903ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}P(3)&=3^{2}-1=8,\\P(X^{2}+1)&=\left(X^{2}+1\right)^{2}-1=X^{4}+2X^{2}\end{aligned}}}"></dd></dl> (in the first example R = K, and in the second one R = K[X]). Substituting X for itself results in <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=P(X),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=P(X),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf0dded030cff79ca51db34ecdde372f3b5d283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.025ex; height:2.843ex;" alt="{\displaystyle P=P(X),}"></dd></dl> explaining why the sentences "Let P be a polynomial" and "Let P(X) be a polynomial" are equivalent. The polynomial function defined by a polynomial P is the function from K into K that is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto P(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto P(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf63b200afc3d9b7cb993fbc51c8cf051f957f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.475ex; height:2.843ex;" alt="{\displaystyle x\mapsto P(x).}"> If K is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if K is a field with q elements, then the polynomials 0 and Xq − X both define the zero function. For every a in R, the evaluation at a, that is, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\mapsto P(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↦</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\mapsto P(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75ca1466229080cda9212a275e518277ee2a49b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.144ex; height:2.843ex;" alt="{\displaystyle P\mapsto P(a)}"> defines an <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> from K[X] to R, which is the unique homomorphism from K[X] to R that fixes K, and maps X to a. In other words, K[X] has the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>: <dl><dd>For every ring R containing K, and every element a of R, there is a unique algebra homomorphism from K[X] to R that fixes K, and maps X to a.</dd></dl> As for all universal properties, this defines the pair (K[X], X) up to a unique isomorphism, and can therefore be taken as a definition of K[X]. The <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\mapsto P(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">↦</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\mapsto P(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a75ca1466229080cda9212a275e518277ee2a49b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.144ex; height:2.843ex;" alt="{\displaystyle P\mapsto P(a)}">, that is, the subset of R obtained by substituting a for X in elements of K[X], is denoted K[a].<a href="#cite_note-14">[14]</a> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{P({\sqrt {2}})\mid P(X)\in \mathbb {Z} [X]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{P({\sqrt {2}})\mid P(X)\in \mathbb {Z} [X]\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae2cec3ca445fa760df51ba17d48477fcb6c729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.155ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{P({\sqrt {2}})\mid P(X)\in \mathbb {Z} [X]\}}">, and the simplification rules for the powers of a square root imply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{a+b{\sqrt {2}}\mid a\in \mathbb {Z} ,b\in \mathbb {Z} \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>∣</mo> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{a+b{\sqrt {2}}\mid a\in \mathbb {Z} ,b\in \mathbb {Z} \}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f00ab5c4c213c36a225bb48279b0dfe87f2be34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.159ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [{\sqrt {2}}]=\{a+b{\sqrt {2}}\mid a\in \mathbb {Z} ,b\in \mathbb {Z} \}.}"> <div class="mw-heading mw-heading2"><h2 id="Univariate_polynomials_over_a_field">Univariate polynomials over a field</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=4" title="Edit section: Univariate polynomials over a field">edit</a>]</div> If K is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, the polynomial ring K[X] has many properties that are similar to those of the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}"> Most of these similarities result from the similarity between the <a href="/wiki/Long_division" title="Long division">long division of integers</a> and the <a href="/wiki/Polynomial_long_division" title="Polynomial long division">long division of polynomials</a>. Most of the properties of K[X] that are listed in this section do not remain true if K is not a field, or if one considers polynomials in several indeterminates. Like for integers, the <a href="/wiki/Euclidean_division_of_polynomials" class="mw-redirect" title="Euclidean division of polynomials">Euclidean division of polynomials</a> has a property of uniqueness. That is, given two polynomials a and b ≠ 0 in K[X], there is a unique pair (q, r) of polynomials such that a = bq + r, and either r = 0 or deg(r) < deg(b). This makes K[X] a <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a>. However, most other Euclidean domains (except integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division. The Euclidean division is the basis of the <a href="/wiki/Euclidean_algorithm_for_polynomials" class="mw-redirect" title="Euclidean algorithm for polynomials">Euclidean algorithm for polynomials</a> that computes a <a href="/wiki/Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">polynomial greatest common divisor</a> of two polynomials. Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the <a href="/wiki/Preorder" title="Preorder">preorder</a> defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of a and b are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to 1). The <a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a> allows computing (and proving) <a href="/wiki/B%C3%A9zout%27s_identity" title="Bézout's identity">Bézout's identity</a>. In the case of K[X], it may be stated as follows. Given two polynomials p and q of respective degrees m and n, if their monic greatest common divisor g has the degree d, then there is a unique pair (a, b) of polynomials such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ap+bq=g,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>p</mi> <mo>+</mo> <mi>b</mi> <mi>q</mi> <mo>=</mo> <mi>g</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ap+bq=g,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92a84e9630875a9835660aaf79cd12489630fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.168ex; height:2.509ex;" alt="{\displaystyle ap+bq=g,}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(a)\leq n-d,\quad \deg(b)<m-d.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mi>d</mi> <mo>,</mo> <mspace width="1em" /> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>m</mi> <mo>−</mo> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(a)\leq n-d,\quad \deg(b)<m-d.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcf14ca3771d0fca89bf73f367c084213b8acdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.569ex; height:2.843ex;" alt="{\displaystyle \deg(a)\leq n-d,\quad \deg(b)<m-d.}"></dd></dl> (For making this true in the limiting case where m = d or n = d, one has to define as negative the degree of the zero polynomial. Moreover, the equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(a)=n-d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(a)=n-d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7f803022b2b0220c8d7d9351563ddbfb49a1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.076ex; height:2.843ex;" alt="{\displaystyle \deg(a)=n-d}"> can occur only if p and q are associated.) The uniqueness property is rather specific to K[X]. In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require a > 0. <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a> applies to K[X]. That is, if a divides bc, and is <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> with b, then a divides c. Here, coprime means that the monic greatest common divisor is 1. Proof: By hypothesis and Bézout's identity, there are e, p, and q such that ae = bc and 1 = ap + bq. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>p</mi> <mo>+</mo> <mi>b</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>a</mi> <mi>p</mi> <mo>+</mo> <mi>a</mi> <mi>e</mi> <mi>q</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mi>p</mi> <mo>+</mo> <mi>e</mi> <mi>q</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a6c85cd218f44e9a59a4fc3f6833193d63a8cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.91ex; height:2.843ex;" alt="{\displaystyle c=c(ap+bq)=cap+aeq=a(cp+eq).}"> The <a href="/wiki/Unique_factorization" class="mw-redirect" title="Unique factorization">unique factorization</a> property results from Euclid's lemma. In the case of integers, this is the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. In the case of K[X], it may be stated as: every non-constant polynomial can be expressed in a unique way as the product of a constant, and one or several irreducible monic polynomials; this decomposition is unique up to the order of the factors. In other terms K[X] is a <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>. If K is the field of complex numbers, the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> asserts that a univariate polynomial is irreducible if and only if its degree is one. In this case the unique factorization property can be restated as: every non-constant univariate polynomial over the complex numbers can be expressed in a unique way as the product of a constant, and one or several polynomials of the form X − r; this decomposition is unique up to the order of the factors. For each factor, r is a <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> of the polynomial, and the number of occurrences of a factor is the <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a> of the corresponding root. <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=5" title="Edit section: Derivation">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup plainlinks metadata ambox ambox-style ambox-Cleanup" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section may require <a href="/wiki/Wikipedia:Cleanup" title="Wikipedia:Cleanup">cleanup</a> to meet Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>. The specific problem is: the subsections reduced to a wikilink require a summary of the linked article. Please help <a href="/wiki/Special:EditPage/Polynomial_ring" title="Special:EditPage/Polynomial ring">improve this section</a> if you can. (June 2023) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Formal_derivative" title="Formal derivative">Formal derivative</a> and <a href="/wiki/Derivation_(differential_algebra)" title="Derivation (differential algebra)">Derivation (differential algebra)</a></div> The <a href="/wiki/Formal_derivative" title="Formal derivative">(formal) derivative</a> of the polynomial <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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>2</mn> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d5ad0928f318b0c3fa0681a9ec7d302d48b72a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.632ex; height:3.009ex;" alt="{\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}}"></dd></dl> is the polynomial <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+2a_{2}X+\cdots +na_{n}X^{n-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>X</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <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> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+2a_{2}X+\cdots +na_{n}X^{n-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc369d74d6b6261ed4157aa8d868885495ca42c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.761ex; height:3.009ex;" alt="{\displaystyle a_{1}+2a_{2}X+\cdots +na_{n}X^{n-1}.}"></dd></dl> In the case of polynomials with <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> coefficients, this is the standard <a href="/wiki/Derivative" title="Derivative">derivative</a>. The above formula defines the derivative of a polynomial even if the coefficients belong to a ring on which no notion of <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> is defined. The derivative makes the polynomial ring a <a href="/wiki/Differential_algebra" title="Differential algebra">differential algebra</a>. The existence of the derivative is one of the main properties of a polynomial ring that is not shared with integers, and makes some computations easier on a polynomial ring than on integers. <div class="mw-heading mw-heading4"><h4 id="Square-free_factorization">Square-free factorization</h4>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=6" title="Edit section: Square-free factorization">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Square-free_polynomial" title="Square-free polynomial">Square-free polynomial</a></div> <div class="mw-heading mw-heading4"><h4 id="Lagrange_interpolation">Lagrange interpolation</h4>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=7" title="Edit section: Lagrange interpolation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lagrange_polynomial#Barycentric_form" title="Lagrange polynomial">Lagrange polynomial § Barycentric form</a></div> <div class="mw-heading mw-heading4"><h4 id="Polynomial_decomposition">Polynomial decomposition</h4>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=8" title="Edit section: Polynomial decomposition">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Polynomial_decomposition" title="Polynomial decomposition">Polynomial decomposition</a></div> <div class="mw-heading mw-heading3"><h3 id="Factorization">Factorization</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=9" title="Edit section: Factorization">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">Polynomial factorization</a></div> Except for factorization, all previous properties of K[X] are <a href="/wiki/Effective_proof" class="mw-redirect" title="Effective proof">effective</a>, since their proofs, as sketched above, are associated with <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a> is a <a href="/wiki/Quadratic_time" class="mw-redirect" title="Quadratic time">quadratic</a> function of the input size. The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing. Already for the integers, there is no known algorithm running on a classical (non-quantum) computer for factorizing them in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>. This is the basis of the <a href="/wiki/RSA_cryptosystem" class="mw-redirect" title="RSA cryptosystem">RSA cryptosystem</a>, widely used for secure Internet communications. In the case of K[X], the factors, and the methods for computing them, depend strongly on K. Over the complex numbers, the irreducible factors (those that cannot be factorized further) are all of degree one, while, over the real numbers, there are irreducible polynomials of degree 2, and, over the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, there are irreducible polynomials of any degree. For example, the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{4}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{4}-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a794db903a72742c4800a0b225c165417c622af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle X^{4}-2}"> is irreducible over the rational numbers, is factored as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X^{2}+{\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X^{2}+{\sqrt {2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7388b78e8360ee5ea6bb642569e3ff42dcf7f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.255ex; height:3.176ex;" alt="{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X^{2}+{\sqrt {2}})}"> over the real numbers and, and as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90eb19786144d561600d00a4ffa682c257467e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.517ex; height:3.176ex;" alt="{\displaystyle (X-{\sqrt[{4}]{2}})(X+{\sqrt[{4}]{2}})(X-i{\sqrt[{4}]{2}})(X+i{\sqrt[{4}]{2}})}"> over the complex numbers. The existence of a factorization algorithm depends also on the ground field. In the case of the real or complex numbers, <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> shows that the roots of some polynomials, and thus the irreducible factors, cannot be computed exactly. Therefore, a factorization algorithm can compute only approximations of the factors. Various algorithms have been designed for computing such approximations, see <a href="/wiki/Root_finding_of_polynomials" class="mw-redirect" title="Root finding of polynomials">Root finding of polynomials</a>. There is an example of a field K such that there exist exact algorithms for the arithmetic operations of K, but there cannot exist any algorithm for deciding whether a polynomial of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{p}-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{p}-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c89727242046a8131f11cd9925bb3b32903a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.126ex; height:2.509ex;" alt="{\displaystyle X^{p}-a}"> is <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> or is a product of polynomials of lower degree.<a href="#cite_note-15">[15]</a> On the other hand, over the rational numbers and over finite fields, the situation is better than for <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a>, as there are <a href="/wiki/Factorization_of_polynomials" title="Factorization of polynomials">factorization algorithms</a> that have a <a href="/wiki/Polynomial_complexity" class="mw-redirect" title="Polynomial complexity">polynomial complexity</a>. They are implemented in most general purpose <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>. <div class="mw-heading mw-heading3"><h3 id="Minimal_polynomial">Minimal polynomial</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=10" title="Edit section: Minimal polynomial">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">Minimal polynomial (field theory)</a></div> If θ is an element of an <a href="/wiki/Associative_algebra" title="Associative algebra">associative K-algebra</a> L, the <a href="#Polynomial_evaluation">polynomial evaluation</a> at θ is the unique <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> φ from K[X] into L that maps X to θ and does not affect the elements of K itself (it is the <a href="/wiki/Identity_function" title="Identity function">identity map</a> on K). It consists of substituting X with θ in every polynomial. That is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \left(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}\right)=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <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> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>θ</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \left(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}\right)=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635eb6ad95e5d52ecb84dc69b026cf78ebb147c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:81.055ex; height:3.343ex;" alt="{\displaystyle \varphi \left(a_{m}X^{m}+a_{m-1}X^{m-1}+\cdots +a_{1}X+a_{0}\right)=a_{m}\theta ^{m}+a_{m-1}\theta ^{m-1}+\cdots +a_{1}\theta +a_{0}.}"></dd></dl> The image of this evaluation homomorphism is the subalgebra generated by θ, which is necessarily commutative. If φ is injective, the subalgebra generated by θ is isomorphic to K[X]. In this case, this subalgebra is often denoted by K[θ]. The notation ambiguity is generally harmless, because of the isomorphism. If the evaluation homomorphism is not injective, this means that its <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> is a nonzero <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a>, consisting of all polynomials that become zero when X is substituted with θ. This ideal consists of all multiples of some monic polynomial, that is called the minimal polynomial of θ. The term minimal is motivated by the fact that its degree is minimal among the degrees of the elements of the ideal. There are two main cases where minimal polynomials are considered. In <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a> and <a href="/wiki/Number_theory" title="Number theory">number theory</a>, an element θ of an <a href="/wiki/Extension_field" class="mw-redirect" title="Extension field">extension field</a> L of K is <a href="/wiki/Algebraic_element" title="Algebraic element">algebraic</a> over K if it is a root of some polynomial with coefficients in K. The <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the <a href="/wiki/Complex_number" title="Complex number">complex number</a> i is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759c679330a1c67db74a3da9ee5cca488de3a589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle X^{2}+1}">. The <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomials</a> are the minimal polynomials of the <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>. In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, the n×n <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a> over K form an <a href="/wiki/Associative_algebra" title="Associative algebra">associative K-algebra</a> of finite dimension (as a vector space). Therefore the evaluation homomorphism cannot be injective, and every matrix has a <a href="/wiki/Minimal_polynomial_(linear_algebra)" title="Minimal polynomial (linear algebra)">minimal polynomial</a> (not necessarily irreducible). By <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a>, the evaluation homomorphism maps to zero the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most n. <div class="mw-heading mw-heading3"><h3 id="Quotient_ring">Quotient ring</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=11" title="Edit section: Quotient ring">edit</a>]</div> In the case of K[X], the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> by an ideal can be built, as in the general case, as a set of <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a>. However, as each equivalence class contains exactly one polynomial of minimal degree, another construction is often more convenient. Given a polynomial p of degree d, the quotient ring of K[X] by the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by p can be identified with the <a href="/wiki/Vector_space" title="Vector space">vector space</a> of the polynomials of degrees less than d, with the "multiplication modulo p" as a multiplication, the multiplication modulo p consisting of the remainder under the division by p of the (usual) product of polynomials. This quotient ring is variously denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/pK[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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/pK[X],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e52be91379b318a4f22919604ec7f8d9f2d1e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.658ex; height:2.843ex;" alt="{\displaystyle K[X]/pK[X],}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/\langle p\rangle ,}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/\langle p\rangle ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/337bc88869134a1aeba2c8700c0b6a60aafdd362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.128ex; height:2.843ex;" alt="{\displaystyle K[X]/\langle p\rangle ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p),}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/(p),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a047c03e08df13a7edb8c084cac66aca071e8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.128ex; height:2.843ex;" alt="{\displaystyle K[X]/(p),}"> or simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/p.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a2a61ff184564b687382271d277951d09cf149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.318ex; height:2.843ex;" alt="{\displaystyle K[X]/p.}"> The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p)}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/(p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf01b3af7443f2d6228b55d165abf44f9febf70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.481ex; height:2.843ex;" alt="{\displaystyle K[X]/(p)}"> is a field if and only if p is an <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a>. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p, and <a href="/wiki/B%C3%A9zout%27s_identity" title="Bézout's identity">Bézout's identity</a> allows computing r and s such that sp + qr = 1; so, r is the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of q modulo p. Conversely, if p is reducible, then there exist polynomials a, b of degrees lower than deg(p) such that ab = p ; so a, b are nonzero <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a> modulo p, and cannot be invertible. For example, the standard definition of the field of the complex numbers can be summarized by saying that it is the quotient ring <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} =\mathbb {R} [X]/(X^{2}+1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} =\mathbb {R} [X]/(X^{2}+1),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7203506bc7eaacd73467ad4b95bd0cdb9db0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.401ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} =\mathbb {R} [X]/(X^{2}+1),}"></dd></dl> and that the image of X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> is denoted by i. In fact, by the above description, this quotient consists of all polynomials of degree one in i, which have the form a + bi, with a and b in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} .}"> The remainder of the Euclidean division that is needed for multiplying two elements of the quotient ring is obtained by replacing i2 by −1 in their product as polynomials (this is exactly the usual definition of the product of complex numbers). Let θ be an <a href="/wiki/Algebraic_element" title="Algebraic element">algebraic element</a> in a K-algebra A. By algebraic, one means that θ has a minimal polynomial p. The <a href="/wiki/First_ring_isomorphism_theorem" class="mw-redirect" title="First ring isomorphism theorem">first ring isomorphism theorem</a> asserts that the substitution homomorphism induces an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p)}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/(p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf01b3af7443f2d6228b55d165abf44f9febf70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.481ex; height:2.843ex;" alt="{\displaystyle K[X]/(p)}"> onto the image K[θ] of the substitution homomorphism. In particular, if A is a <a href="/wiki/Simple_extension" title="Simple extension">simple extension</a> of K generated by θ, this allows identifying A and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/(p).}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/(p).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c8230122367e1a79b8989274fa065b453332d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.128ex; height:2.843ex;" alt="{\displaystyle K[X]/(p).}"> This identification is widely used in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. <div class="mw-heading mw-heading3"><h3 id="Modules">Modules</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=12" title="Edit section: Modules">edit</a>]</div> The <a href="/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain" title="Structure theorem for finitely generated modules over a principal ideal domain">structure theorem for finitely generated modules over a principal ideal domain</a> applies to K[X], when K is a field. This means that every finitely generated module over K[X] may be decomposed into a <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a> of a <a href="/wiki/Free_module" title="Free module">free module</a> and finitely many modules of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X]/\left\langle P^{k}\right\rangle }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow> <mo>⟨</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X]/\left\langle P^{k}\right\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6693e2e2afd887e4df88c3d377ef760d3af3033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.994ex; height:3.343ex;" alt="{\displaystyle K[X]/\left\langle P^{k}\right\rangle }">, where P is an <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a> over K and k a positive integer. <div class="mw-heading mw-heading2"><h2 id="Definition_(multivariate_case)">Definition (multivariate case)</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=13" title="Edit section: Definition (multivariate case)">edit</a>]</div> Given n symbols <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d11af8d1db16f154c91c2bff262f719a8e43f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.946ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n},}"> called <a href="/wiki/Indeterminate_(variable)" title="Indeterminate (variable)">indeterminates</a>, a <a href="/wiki/Monomial" title="Monomial">monomial</a> (also called power product) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea9a48bbf309c977a1900d005f9b26f424b5e5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.856ex; height:3.176ex;" alt="{\displaystyle X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}"></dd></dl> is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}^{0}\cdots X_{n}^{0}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}^{0}\cdots X_{n}^{0}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c553c3ed9f3278f8af33b36b5cd2df5ff5ff225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.599ex; height:3.176ex;" alt="{\displaystyle X_{1}^{0}\cdots X_{n}^{0}=1.}"> The <a href="/wiki/Tuple" title="Tuple">tuple</a> of exponents α = (α1, …, αn) is called the multidegree or exponent vector of the monomial. For a less cumbersome notation, the abbreviation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\alpha }=X_{1}^{\alpha _{1}}\cdots 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> <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 }=X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb41cf1e16a0cc45f2454ef4216dab0247063bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.235ex; height:3.176ex;" alt="{\displaystyle X^{\alpha }=X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}}"></dd></dl> is often used. The degree of a monomial Xα, frequently denoted deg α or |α|, is the sum of its exponents: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg \alpha =\sum _{i=1}^{n}\alpha _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg \alpha =\sum _{i=1}^{n}\alpha _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9aed85e58041c58d9532682fb054a554df632d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.137ex; height:6.843ex;" alt="{\displaystyle \deg \alpha =\sum _{i=1}^{n}\alpha _{i}.}"></dd></dl> A polynomial in these indeterminates, with coefficients in a field K, or more generally a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, is a finite <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of monomials <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\sum _{\alpha }p_{\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>p</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\sum _{\alpha }p_{\alpha }X^{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7365ac9856ca58fa7ad50fbb59b9646f3b043541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:13.834ex; height:5.509ex;" alt="{\displaystyle p=\sum _{\alpha }p_{\alpha }X^{\alpha }}"></dd></dl> with coefficients in K. The degree of a nonzero polynomial is the maximum of the degrees of its monomials with nonzero coefficients. The set of polynomials in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d11af8d1db16f154c91c2bff262f719a8e43f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.946ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n},}"> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11a7e9bbac4ce6c4970f823182e7580f0299e952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.306ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}],}"> is thus a <a href="/wiki/Vector_space" title="Vector space">vector space</a> (or a <a href="/wiki/Free_module" title="Free module">free module</a>, if K is a ring) that has the monomials as a basis. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cf0764b4682f6c7a4827df7693feccd72ff86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}]}"> is naturally equipped (see below) with a multiplication that makes a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> over K, called the polynomial ring in n indeterminates over K (the definite article the reflects that it is uniquely defined up to the name and the order of the indeterminates. If the ring K is <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cf0764b4682f6c7a4827df7693feccd72ff86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}]}"> is also a commutative ring. <div class="mw-heading mw-heading3"><h3 id="Operations_in_K[X1,_...,_Xn]">Operations in K[X1, ..., Xn]</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=14" title="Edit section: Operations in K[X1, ..., Xn]">edit</a>]</div> Addition and scalar multiplication of polynomials are those of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> or <a href="/wiki/Free_module" title="Free module">free module</a> equipped by a specific basis (here the basis of the monomials). Explicitly, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },}"> <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> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>p</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> <mspace width="1em" /> <mi>q</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b42c5122dacb316e5c124daa9ec144b1ac3ea8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.089ex; width:31.13ex; height:5.843ex;" alt="{\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },}"> where I and J are finite sets of exponent vectors. The scalar multiplication of p and a scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e894da36b4db8c32e35962d3950750ad556f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle c\in K}"> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cp=\sum _{\alpha \in I}cp_{\alpha }X^{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>p</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>c</mi> <msub> <mi>p</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 cp=\sum _{\alpha \in I}cp_{\alpha }X^{\alpha }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35593d29ee114427d3e0edd57a138158b3e3f5d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.405ex; height:5.676ex;" alt="{\displaystyle cp=\sum _{\alpha \in I}cp_{\alpha }X^{\alpha }.}"></dd></dl> The addition of p and q is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>I</mi> <mo>∪</mo> <mi>J</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <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+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d65b3e661459dc2f908a0a273fd499275dbfd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:26.733ex; height:5.676ex;" alt="{\displaystyle p+q=\sum _{\alpha \in I\cup J}(p_{\alpha }+q_{\alpha })X^{\alpha },}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\alpha }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\alpha }=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2139b4e777b292ab16a4e62c6de185455ee5ea2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.804ex; height:2.509ex;" alt="{\displaystyle p_{\alpha }=0}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \not \in I,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∉</mo> <mi>I</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \not \in I,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a89c2ce3f55baffe30e9cb337573faca198822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.147ex; height:2.676ex;" alt="{\displaystyle \alpha \not \in I,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\beta }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\beta }=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4cc03332c54e0b96f2d8ea6cf41877d49c3b204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.472ex; height:2.843ex;" alt="{\displaystyle q_{\beta }=0}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \not \in J.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>∉</mo> <mi>J</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \not \in J.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36508f9f7c96ffc2cc6363277ff6c42f4a0c5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.676ex;" alt="{\displaystyle \beta \not \in J.}"> Moreover, if one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\alpha }+q_{\alpha }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\alpha }+q_{\alpha }=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83430a9c4c701a278bae46ef2ae8f678d3423e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.966ex; height:2.509ex;" alt="{\displaystyle p_{\alpha }+q_{\alpha }=0}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in I\cap J,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mi>I</mi> <mo>∩</mo> <mi>J</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in I\cap J,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61388279ee3615b096e1308a624539ce723f1647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.201ex; height:2.509ex;" alt="{\displaystyle \alpha \in I\cap J,}"> the corresponding zero term is removed from the result. The multiplication is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }p_{\alpha }q_{\beta }\right)X^{\gamma },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>q</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mo>∈</mo> <mi>I</mi> <mo>+</mo> <mi>J</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∣</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <mi>γ</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }p_{\alpha }q_{\beta }\right)X^{\gamma },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7dc88dfeec750181bcf03833079695e5774efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-left: -0.089ex; width:32.577ex; height:8.176ex;" alt="{\displaystyle pq=\sum _{\gamma \in I+J}\left(\sum _{\alpha ,\beta \mid \alpha +\beta =\gamma }p_{\alpha }q_{\beta }\right)X^{\gamma },}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I+J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>+</mo> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I+J}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0e36f3880c3eae9b440c7e412a0f02664b8624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.484ex; height:2.343ex;" alt="{\displaystyle I+J}"> is the set of the sums of one exponent vector in I and one other in J (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the factors. The verification of the axioms of an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> is straightforward. <div class="mw-heading mw-heading3"><h3 id="Polynomial_expression">Polynomial expression</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=15" title="Edit section: Polynomial expression">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_expression" title="Algebraic expression">Algebraic expression</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Polynomial_ring" title="Special:EditPage/Polynomial ring">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (January 2021) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> A polynomial expression is an <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> built with scalars (elements of K), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers. As all these operations are defined in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cf0764b4682f6c7a4827df7693feccd72ff86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}]}"> a polynomial expression represents a polynomial, that is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\dots ,X_{n}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\dots ,X_{n}].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5971b61b083b8320deec96bd98af348bb33c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.306ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\dots ,X_{n}].}"> The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial. Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a> all the products that have a sum among their factors, and then using <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> (except for the product of two scalars), and <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the <a href="/wiki/Like_terms" title="Like terms">like terms</a>. The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a>, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation. <div class="mw-heading mw-heading3"><h3 id="Categorical_characterization">Categorical characterization</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=16" title="Edit section: Categorical characterization">edit</a>]</div> If K is a commutative ring, the polynomial ring K[X1, …, Xn] has the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>: for every <a href="/wiki/Commutative_algebra_(structure)" class="mw-redirect" title="Commutative algebra (structure)">commutative K-algebra</a> A, and every n-<a href="/wiki/Tuple" title="Tuple">tuple</a> (x1, …, xn) of elements of A, there is a unique <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> from K[X1, …, Xn] to A that maps each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> to the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca686849ad350b47ebab4a394b66d84d5fa942bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.009ex;" alt="{\displaystyle x_{i}.}"> This homomorphism is the evaluation homomorphism that consists in substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"> in every polynomial. As it is the case for every universal property, this characterizes the pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa09822e563058b767425df5abf0097dffe1b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.611ex; height:2.843ex;" alt="{\displaystyle (K[X_{1},\dots ,X_{n}],(X_{1},\dots ,X_{n}))}"> up to a unique <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. This may also be interpreted in terms of <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint functors</a>. More precisely, let SET and ALG be respectively the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a> of sets and commutative K-algebras (here, and in the following, the morphisms are trivially defined). There is a <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">G</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e37f477291831ab140c086d3d7ba63e73eee6629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.643ex; height:2.176ex;" alt="{\displaystyle \mathrm {F} :\mathrm {ALG} \to \mathrm {SET} }"> that maps algebras to their underlying sets. On the other hand, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mapsto K[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">↦</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mapsto K[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56dfb938869f70cb50f04d3ec6e66fceebe3b9d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.934ex; height:2.843ex;" alt="{\displaystyle X\mapsto K[X]}"> defines a functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">L</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">G</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f76a3cd36649e049d78f5f730aa4bc09ada0189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.969ex; height:2.176ex;" alt="{\displaystyle \mathrm {POL} :\mathrm {SET} \to \mathrm {ALG} }"> in the other direction. (If X is infinite, K[X] is the set of all polynomials in a finite number of elements of X.) The universal property of the polynomial ring means that F and POL are <a href="/wiki/Adjoint_functors" title="Adjoint functors">adjoint functors</a>. That is, there is a bijection <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{\mathrm {SET} }(X,\operatorname {F} (A))\cong \operatorname {Hom} _{\mathrm {ALG} }(K[X],A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">F</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≅</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">G</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{\mathrm {SET} }(X,\operatorname {F} (A))\cong \operatorname {Hom} _{\mathrm {ALG} }(K[X],A).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3587f518c9c731771ac7fbea5b3aefcfa747ad1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.482ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{\mathrm {SET} }(X,\operatorname {F} (A))\cong \operatorname {Hom} _{\mathrm {ALG} }(K[X],A).}"></dd></dl> This may be expressed also by saying that polynomial rings are free commutative algebras, since they are <a href="/wiki/Free_object" title="Free object">free objects</a> in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients is the free commutative ring over its set of variables, since commutative rings and commutative algebras over the integers are the same thing. <div class="mw-heading mw-heading2"><h2 id="Graded_structure">Graded structure</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=17" title="Edit section: Graded structure">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Empty_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></td><td class="mbox-text"><div class="mbox-text-span">This section is empty. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Polynomial_ring&action=edit&section=">adding to it</a>. (April 2022)</div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Univariate_over_a_ring_vs._multivariate">Univariate over a ring vs. multivariate</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=18" title="Edit section: Univariate over a ring vs. multivariate">edit</a>]</div> A polynomial in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> can be considered as a univariate polynomial in the indeterminate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"> over the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n-1}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n-1}],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726494480638e29f49e3999e1f5c2d4d99d51933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.406ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n-1}],}"> by regrouping the terms that contain the same power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6759dac8dbc82523fd14db264c829247879174b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.79ex; height:2.509ex;" alt="{\displaystyle X_{n},}"> that is, by using the identity <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{(\alpha _{1},\ldots ,\alpha _{n})\in I}c_{\alpha _{1},\ldots ,\alpha _{n}}X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}=\sum _{i}\left(\sum _{(\alpha _{1},\ldots ,\alpha _{n-1})\mid (\alpha _{1},\ldots ,\alpha _{n-1},i)\in I}c_{\alpha _{1},\ldots ,\alpha _{n-1}}X_{1}^{\alpha _{1}}\cdots X_{n-1}^{\alpha _{n-1}}\right)X_{n}^{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <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> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <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> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <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> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{(\alpha _{1},\ldots ,\alpha _{n})\in I}c_{\alpha _{1},\ldots ,\alpha _{n}}X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}=\sum _{i}\left(\sum _{(\alpha _{1},\ldots ,\alpha _{n-1})\mid (\alpha _{1},\ldots ,\alpha _{n-1},i)\in I}c_{\alpha _{1},\ldots ,\alpha _{n-1}}X_{1}^{\alpha _{1}}\cdots X_{n-1}^{\alpha _{n-1}}\right)X_{n}^{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6c95b27250e5a5cf12475486e1aae8a5b84b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:91.743ex; height:8.343ex;" alt="{\displaystyle \sum _{(\alpha _{1},\ldots ,\alpha _{n})\in I}c_{\alpha _{1},\ldots ,\alpha _{n}}X_{1}^{\alpha _{1}}\cdots X_{n}^{\alpha _{n}}=\sum _{i}\left(\sum _{(\alpha _{1},\ldots ,\alpha _{n-1})\mid (\alpha _{1},\ldots ,\alpha _{n-1},i)\in I}c_{\alpha _{1},\ldots ,\alpha _{n-1}}X_{1}^{\alpha _{1}}\cdots X_{n-1}^{\alpha _{n-1}}\right)X_{n}^{i},}"></dd></dl> which results from the distributivity and associativity of ring operations. This means that one has an <a href="/wiki/Algebra_isomorphism" class="mw-redirect" title="Algebra isomorphism">algebra isomorphism</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]\cong (K[X_{1},\ldots ,X_{n-1}])[X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≅</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]\cong (K[X_{1},\ldots ,X_{n-1}])[X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c3ee9af59d2086949741d7059b6a2f3ddcb4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.763ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]\cong (K[X_{1},\ldots ,X_{n-1}])[X_{n}]}"></dd></dl> that maps each indeterminate to itself. (This isomorphism is often written as an equality, which is justified by the fact that polynomial rings are defined up to a unique isomorphism.) In other words, a multivariate polynomial ring can be considered as a univariate polynomial over a smaller polynomial ring. This is commonly used for proving properties of multivariate polynomial rings, by <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a> on the number of indeterminates. The main such properties are listed below. <div class="mw-heading mw-heading3"><h3 id="Properties_that_pass_from_R_to_R[X]">Properties that pass from R to R[X]</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=19" title="Edit section: Properties that pass from R to R[X]">edit</a>]</div> In this section, R is a commutative ring, K is a field, X denotes a single indeterminate, and, as usual, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> is the ring of integers. Here is the list of the main ring properties that remain true when passing from R to R[X]. <ul><li>If R is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> then the same holds for R[X] (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors). <ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d22cfd2dfdbfee457e462397218f0845d1d5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> are integral domains.</li></ul></li> <li>If R is a <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a> then the same holds for R[X]. This results from <a href="/wiki/Gauss%27s_lemma_(polynomial)" class="mw-redirect" title="Gauss's lemma (polynomial)">Gauss's lemma</a> and the unique factorization property of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L[X],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L[X],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b467e0e9a96b2086bcee9943ab935fd960491a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle L[X],}"> where L is the field of fractions of R. <ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d22cfd2dfdbfee457e462397218f0845d1d5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> are unique factorization domains.</li></ul></li> <li>If R is a <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian ring</a>, then the same holds for R[X]. <ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d22cfd2dfdbfee457e462397218f0845d1d5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> are Noetherian rings; this is <a href="/wiki/Hilbert%27s_basis_theorem" title="Hilbert's basis theorem">Hilbert's basis theorem</a>.</li></ul></li> <li>If R is a Noetherian ring, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim R[X]=1+\dim R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim R[X]=1+\dim R,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4262e007a0121c85eb2ae8f6e3a2165b55691b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.075ex; height:2.843ex;" alt="{\displaystyle \dim R[X]=1+\dim R,}"> where "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66115c83c4bb19068adb45849f0f596647c18f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \dim }">" denotes the <a href="/wiki/Krull_dimension" title="Krull dimension">Krull dimension</a>. <ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a255aafe61b6fcedfdc73a8e0e7368c5ad0129c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.415ex; height:2.843ex;" alt="{\displaystyle \dim K[X_{1},\ldots ,X_{n}]=n}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646107e48d7f66b4426d2792c33a718cefcbae19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.549ex; height:2.843ex;" alt="{\displaystyle \dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1.}"></li></ul></li> <li>If R is a <a href="/wiki/Regular_ring" class="mw-redirect" title="Regular ring">regular ring</a>, then the same holds for R[X]; in this case, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim R[X]=\dim R[X]=1+\operatorname {gl} \,\dim R=1+\dim R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gl</mi> <mspace width="thinmathspace" /> <mi>dim</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>gl</mi> <mspace width="thinmathspace" /> <mi>dim</mi> <mo>⁡</mo> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim R[X]=\dim R[X]=1+\operatorname {gl} \,\dim R=1+\dim R,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc31f932f909288aa48bcc56407e1b1f3e335b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.768ex; height:2.843ex;" alt="{\displaystyle \operatorname {gl} \,\dim R[X]=\dim R[X]=1+\operatorname {gl} \,\dim R=1+\dim R,}"> where "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gl</mi> <mspace width="thinmathspace" /> <mi>dim</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04b60de26f746c6abd1715d4c6eec823b523424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.459ex; height:2.509ex;" alt="{\displaystyle \operatorname {gl} \,\dim }">" denotes the <a href="/wiki/Global_dimension" title="Global dimension">global dimension</a>. <ul><li>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d22cfd2dfdbfee457e462397218f0845d1d5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.143ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},\ldots ,X_{n}]}"> are regular rings, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gl</mi> <mspace width="thinmathspace" /> <mi>dim</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832c2bba7901e808089559e18a8646a239219853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.132ex; height:2.843ex;" alt="{\displaystyle \operatorname {gl} \,\dim \mathbb {Z} [X_{1},\ldots ,X_{n}]=n+1,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gl</mi> <mspace width="thinmathspace" /> <mi>dim</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de3c6e7e07280534e37854b6d8902a999f5f75fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.645ex; height:2.843ex;" alt="{\displaystyle \operatorname {gl} \,\dim K[X_{1},\ldots ,X_{n}]=n.}"> The latter equality is <a href="/wiki/Hilbert%27s_syzygy_theorem" title="Hilbert's syzygy theorem">Hilbert's syzygy theorem</a>.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Several_indeterminates_over_a_field">Several indeterminates over a field</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=20" title="Edit section: Several indeterminates over a field">edit</a>]</div> Polynomial rings in several variables over a field are fundamental in <a href="/wiki/Invariant_theory" title="Invariant theory">invariant theory</a> and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. Some of their properties, such as those described above can be reduced to the case of a single indeterminate, but this is not always the case. In particular, because of the geometric applications, many interesting properties must be invariant under <a href="/wiki/Affine_transformation" title="Affine transformation">affine</a> or <a href="/wiki/Projective_transformation" class="mw-redirect" title="Projective transformation">projective</a> transformations of the indeterminates. This often implies that one cannot select one of the indeterminates for a recurrence on the indeterminates. <a href="/wiki/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a>, <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a> and <a href="/wiki/Jacobian_conjecture" title="Jacobian conjecture">Jacobian conjecture</a> are among the most famous properties that are specific to multivariate polynomials over a field. <div class="mw-heading mw-heading3"><h3 id="Hilbert's_Nullstellensatz">Hilbert's Nullstellensatz</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=21" title="Edit section: Hilbert's Nullstellensatz">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a></div> The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, which extends to the multivariate case some aspects of the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>. It is foundational for <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, as establishing a strong link between the algebraic properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> and the geometric properties of <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a>, that are (roughly speaking) set of points defined by <a href="/wiki/Implicit_equation" class="mw-redirect" title="Implicit equation">implicit polynomial equations</a>. The Nullstellensatz, has three main versions, each being a corollary of any other. Two of these versions are given below. For the third version, the reader is referred to the main article on the Nullstellensatz. The first version generalizes the fact that a nonzero univariate polynomial has a <a href="/wiki/Complex_number" title="Complex number">complex</a> zero if and only if it is not a constant. The statement is: a set of polynomials S in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> has a common zero in an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> containing K, if and only if 1 does not belong to the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by S, that is, if 1 is not a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of elements of S with polynomial coefficients. The second version generalizes the fact that the <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible univariate polynomials</a> over the complex numbers are <a href="/wiki/Associate_elements" class="mw-redirect" title="Associate elements">associate</a> to a polynomial of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-\alpha .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>−</mo> <mi>α</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X-\alpha .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b2f3c4393953f5499cd97fe5bf1a0fcc32e77f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.955ex; height:2.343ex;" alt="{\displaystyle X-\alpha .}"> The statement is: If K is algebraically closed, then the <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideals</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[X_{1},\ldots ,X_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1ac7a7496de3efbb2cd22c42398736c86a2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.659ex; height:2.843ex;" alt="{\displaystyle K[X_{1},\ldots ,X_{n}]}"> have the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4cb14f681920246a23d9ab30e94be08bc512ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.684ex; height:2.843ex;" alt="{\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .}"> <div class="mw-heading mw-heading3"><h3 id="Bézout's_theorem">Bézout's theorem</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=22" title="Edit section: Bézout's theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a></div> Bézout's theorem may be viewed as a multivariate generalization of the version of the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> that asserts that a univariate polynomial of degree n has n complex roots, if they are counted with their multiplicities. In the case of <a href="/wiki/Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">bivariate polynomials</a>, it states that two polynomials of degrees d and e in two variables, which have no common factors of positive degree, have exactly de common zeros in an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> containing the coefficients, if the zeros are counted with their multiplicity and include the <a href="/wiki/Point_at_infinity" title="Point at infinity">zeros at infinity</a>. For stating the general case, and not considering "zero at infinity" as special zeros, it is convenient to work with <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomials</a>, and consider zeros in a <a href="/wiki/Projective_space" title="Projective space">projective space</a>. In this context, a projective zero of a homogeneous polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X_{0},\ldots ,X_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X_{0},\ldots ,X_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa61a5b87352b04e7fa758f42ac9c7bb30de899a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.854ex; height:2.843ex;" alt="{\displaystyle P(X_{0},\ldots ,X_{n})}"> is, up to a scaling, a (n + 1)-<a href="/wiki/Tuple" title="Tuple">tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},\ldots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8811e57cd4d3e4fe4e7e2af3a9822f77746e8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{0},\ldots ,x_{n})}"> of elements of K that is different from (0, …, 0), and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{0},\ldots ,x_{n})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x_{0},\ldots ,x_{n})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78210f993473f3b83a6efb84e5488cd035e3ca1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.926ex; height:2.843ex;" alt="{\displaystyle P(x_{0},\ldots ,x_{n})=0}">. Here, "up to a scaling" means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},\ldots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8811e57cd4d3e4fe4e7e2af3a9822f77746e8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{0},\ldots ,x_{n})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda x_{0},\ldots ,\lambda x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>λ</mi> <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 (\lambda x_{0},\ldots ,\lambda x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de55ec422915084a176d2b9ea56002b81e63e15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.63ex; height:2.843ex;" alt="{\displaystyle (\lambda x_{0},\ldots ,\lambda x_{n})}"> are considered as the same zero for any nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in K.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2edb7ffb8ac74ac2b7b5f4c9bf4108099134aebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.909ex; height:2.176ex;" alt="{\displaystyle \lambda \in K.}"> In other words, a zero is a set of <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> of a point in a projective space of dimension n. Then, Bézout's theorem states: Given n homogeneous polynomials of degrees <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1},\ldots ,d_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1},\ldots ,d_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2b00c84940b9421f7f66f4b7df4c59f3871068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.869ex; height:2.509ex;" alt="{\displaystyle d_{1},\ldots ,d_{n}}"> in n + 1 indeterminates, which have only a finite number of common projective zeros in an <a href="/wiki/Algebraically_closed_extension" class="mw-redirect" title="Algebraically closed extension">algebraically closed extension</a> of K, the sum of the <a href="/wiki/Multiplicity_(mathematics)#Intersection_multipliicty" title="Multiplicity (mathematics)">multiplicities</a> of these zeros is the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}\cdots d_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}\cdots d_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b9071a2af273bee9c725c895ae394ed2b45e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.835ex; height:2.509ex;" alt="{\displaystyle d_{1}\cdots d_{n}.}"> <div class="mw-heading mw-heading3"><h3 id="Jacobian_conjecture">Jacobian conjecture</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=23" title="Edit section: Jacobian conjecture">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Jacobian_conjecture" title="Jacobian conjecture">Jacobian conjecture</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></td><td class="mbox-text"><div class="mbox-text-span">This section needs expansion. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Polynomial_ring&action=edit&section=">adding to it</a>. (June 2020)</div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=24" title="Edit section: Generalizations">edit</a>]</div> Polynomial rings can be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, <a href="/wiki/Noncommutative_polynomial_ring" class="mw-redirect" title="Noncommutative polynomial ring">noncommutative polynomial rings</a>, <a href="/wiki/Skew_polynomial_ring" class="mw-redirect" title="Skew polynomial ring">skew polynomial rings</a>, and polynomial <a href="/wiki/Rig_(mathematics)" class="mw-redirect" title="Rig (mathematics)">rigs</a>. <div class="mw-heading mw-heading3"><h3 id="Infinitely_many_variables">Infinitely many variables</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=25" title="Edit section: Infinitely many variables">edit</a>]</div> One slight generalization of polynomial rings is to allow for infinitely many indeterminates. Each monomial still involves only a finite number of indeterminates (so that its degree remains finite), and each polynomial is a still a (finite) linear combination of monomials. Thus, any individual polynomial involves only finitely many indeterminates, and any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates. This generalization has the same property of usual polynomial rings, of being the <a href="/wiki/Free_commutative_algebra" class="mw-redirect" title="Free commutative algebra">free commutative algebra</a>, the only difference is that it is a <a href="/wiki/Free_object" title="Free object">free object</a> over an infinite set. One can also consider a strictly larger ring, by defining as a generalized polynomial an infinite (or finite) formal sum of monomials with a bounded degree. This ring is larger than the usual polynomial ring, as it includes infinite sums of variables. However, it is smaller than the <a href="/wiki/Power_series_ring#Power_series_in_several_variables" class="mw-redirect" title="Power series ring">ring of power series in infinitely many variables</a>. Such a ring is used for constructing the <a href="/wiki/Ring_of_symmetric_functions" title="Ring of symmetric functions">ring of symmetric functions</a> over an infinite set. <div class="mw-heading mw-heading3"><h3 id="Generalized_exponents">Generalized exponents</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=26" title="Edit section: Generalized exponents">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Monoid_ring" title="Monoid ring">Monoid ring</a></div> A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: Xi ⋅ Xj = Xi+j. A set for which addition makes sense (is closed and associative) is called a <a href="/wiki/Monoid" title="Monoid">monoid</a>. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then cn = an + bn for every n in N. The multiplication is defined as the Cauchy product, so that if c = a ⋅ b, then for each n in N, cn is the sum of all aibj where i, j range over all pairs of elements of N which sum to n. When N is commutative, it is convenient to denote the function a in R[N] as the formal sum: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\in N}a_{n}X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n\in N}a_{n}X^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a018e2f129e085ac7767539c233f8b4dd7076c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.592ex; height:5.676ex;" alt="{\displaystyle \sum _{n\in N}a_{n}X^{n}}"></dd></dl> and then the formulas for addition and multiplication are the familiar: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(a_{n}+b_{n}\right)X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <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> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <msub> <mi>b</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> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(a_{n}+b_{n}\right)X^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec013772477be911637a9633a2f2b75df882693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.099ex; height:7.509ex;" alt="{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)+\left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(a_{n}+b_{n}\right)X^{n}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <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> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <msub> <mi>b</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> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mi>N</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>=</mo> <mi>n</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>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567953e4cbcc5b43c303bc32799dc25d5a9e22e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.417ex; height:7.676ex;" alt="{\displaystyle \left(\sum _{n\in N}a_{n}X^{n}\right)\cdot \left(\sum _{n\in N}b_{n}X^{n}\right)=\sum _{n\in N}\left(\sum _{i+j=n}a_{i}b_{j}\right)X^{n}}"></dd></dl> where the latter sum is taken over all i, j in N that sum to n. Some authors such as (<a href="#CITEREFLang2002">Lang 2002</a>, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers. Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers, (<a href="#CITEREFOsbourne2000">Osbourne 2000</a>, §4.4) harv error: no target: CITEREFOsbourne2000 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>). See also <a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a>. <div class="mw-heading mw-heading3"><h3 id="Power_series">Power series</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=27" title="Edit section: Power series">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></div> Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can also be seen as the <a href="/wiki/Completion_of_a_ring" title="Completion of a ring">ring completion</a> of the polynomial ring with respect to the ideal generated by x. <div class="mw-heading mw-heading3"><h3 id="Noncommutative_polynomial_rings">Noncommutative polynomial rings</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=28" title="Edit section: Noncommutative polynomial rings">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a></div> For polynomial rings of more than one variable, the products X⋅Y and Y⋅X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the <a href="/wiki/Monoid_ring" title="Monoid ring">monoid ring</a> R[N], where the monoid N is the <a href="/wiki/Free_monoid" title="Free monoid">free monoid</a> on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other. Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1. <div class="mw-heading mw-heading3"><h3 id="Differential_and_skew-polynomial_rings">Differential and skew-polynomial rings</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=29" title="Edit section: Differential and skew-polynomial rings">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ore_extension" title="Ore extension">Ore extension</a></div> Other generalizations of polynomials are differential and skew-polynomial rings. A differential polynomial ring is a ring of <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a> formed from a ring R and a <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a> δ of R into R. This derivation operates on R, and will be denoted X, when viewed as an operator. The elements of R also operate on R by multiplication. The <a href="/wiki/Function_composition" title="Function composition">composition of operators</a> is denoted as the usual multiplication. It follows that the relation δ(ab) = aδ(b) + δ(a)b may be rewritten as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\cdot a=a\cdot X+\delta (a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>X</mi> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\cdot a=a\cdot X+\delta (a).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3804c33bb4434385273bfa5a77a86ebea106920b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.451ex; height:2.843ex;" alt="{\displaystyle X\cdot a=a\cdot X+\delta (a).}"></dd></dl> This relation may be extended to define a skew multiplication between two polynomials in X with coefficients in R, which make them a <a href="/wiki/Noncommutative_ring" title="Noncommutative ring">noncommutative ring</a>. The standard example, called a <a href="/wiki/Weyl_algebra" title="Weyl algebra">Weyl algebra</a>, takes R to be a (usual) polynomial ring k[Y ], and δ to be the standard polynomial derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial }{\partial Y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>Y</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial }{\partial Y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9fd20891fa74c1fd9c2d50c62ed85f91e3f6026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.022ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial }{\partial Y}}}">. Taking a = Y in the above relation, one gets the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a>, X⋅Y − Y⋅X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the <a href="/wiki/Weyl_algebra" title="Weyl algebra">Weyl algebra</a>. (<a href="#CITEREFLam2001">Lam 2001</a>, §1,ex1.9). The skew-polynomial ring is defined similarly for a ring R and a ring <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F from the monoid N of the positive integers into the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of R, the formula X n⋅r = F(n)(r)⋅X n allows constructing a skew-polynomial ring. (<a href="#CITEREFLam2001">Lam 2001</a>, §1,ex 1.11) Skew polynomial rings are closely related to <a href="/wiki/Crossed_product" title="Crossed product">crossed product</a> algebras. <div class="mw-heading mw-heading3"><h3 id="Polynomial_rigs">Polynomial rigs</h3>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=30" title="Edit section: Polynomial rigs">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Formal_power_series#On_a_semiring" title="Formal power series">Formal power series § On a semiring</a></div> The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> to the requirement that R only be a <a href="/wiki/Semifield" title="Semifield">semifield</a> or <a href="/wiki/Rig_(mathematics)" class="mw-redirect" title="Rig (mathematics)">rig</a>; the resulting polynomial structure/extension R[X] is a polynomial rig. For example, the set of all multivariate polynomials with <a href="/wiki/Natural_number" title="Natural number">natural number</a> coefficients is a polynomial rig. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=31" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Additive_polynomial" title="Additive polynomial">Additive polynomial</a></li> <li><a href="/wiki/Laurent_polynomial" title="Laurent polynomial">Laurent polynomial</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Polynomial_ring&action=edit&section=32" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1017/9781009457927.012">"Index"</a>, The Art of Legal Problem Solving, Cambridge University Press, pp. 123–126, 2024-03-11, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781009457927.012">10.1017/9781009457927.012</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-009-45792-7" title="Special:BookSources/978-1-009-45792-7"><bdi>978-1-009-45792-7</bdi></a>, retrieved 2024-09-14</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Art+of+Legal+Problem+Solving&rft.atitle=Index&rft.pages=123-126&rft.date=2024-03-11&rft_id=info%3Adoi%2F10.1017%2F9781009457927.012&rft.isbn=978-1-009-45792-7&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1017%2F9781009457927.012&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"> </li> <li id="cite_note-:0-2">^ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/polynomialring">"polynomial ring"</a>. planetmath.org. Retrieved 2024-09-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=planetmath.org&rft.atitle=polynomial+ring&rft_id=https%3A%2F%2Fplanetmath.org%2Fpolynomialring&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://artofproblemsolving.com/wiki/index.php/Polynomial_ring">"Art of Problem Solving"</a>. artofproblemsolving.com. Retrieved 2024-09-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=artofproblemsolving.com&rft.atitle=Art+of+Problem+Solving&rft_id=https%3A%2F%2Fartofproblemsolving.com%2Fwiki%2Findex.php%2FPolynomial_ring&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 153 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> Herstein, Hall p. 73 </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 97 </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 154 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 100 </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAntonBivensDavis2012" class="citation cs2">Anton, Howard; Bivens, Irl C.; Davis, Stephen (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=U2uv84cpJHQC&pg=RA1-PA31">Calculus Single Variable</a>, Wiley, p. 31, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780470647707" title="Special:BookSources/9780470647707"><bdi>9780470647707</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+Single+Variable&rft.pages=31&rft.pub=Wiley&rft.date=2012&rft.isbn=9780470647707&rft.aulast=Anton&rft.aufirst=Howard&rft.au=Bivens%2C+Irl+C.&rft.au=Davis%2C+Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DU2uv84cpJHQC%26pg%3DRA1-PA31&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988">. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSendraWinklerPérez-Diaz2007" class="citation cs2">Sendra, J. Rafael; Winkler, Franz; Pérez-Diaz, Sonia (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=puWxs7KG2D0C&pg=PA250">Rational Algebraic Curves: A Computer Algebra Approach</a>, Algorithms and Computation in Mathematics, vol. 22, Springer, p. 250, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540737247" title="Special:BookSources/9783540737247"><bdi>9783540737247</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rational+Algebraic+Curves%3A+A+Computer+Algebra+Approach&rft.series=Algorithms+and+Computation+in+Mathematics&rft.pages=250&rft.pub=Springer&rft.date=2007&rft.isbn=9783540737247&rft.aulast=Sendra&rft.aufirst=J.+Rafael&rft.au=Winkler%2C+Franz&rft.au=P%C3%A9rez-Diaz%2C+Sonia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpuWxs7KG2D0C%26pg%3DPA250&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988">. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1980" class="citation cs2"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard Whitley</a> (1980), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ayVxeUNbZRAC&pg=PA183">Elementary Matrix Theory</a>, Dover, p. 183, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486150277" title="Special:BookSources/9780486150277"><bdi>9780486150277</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Matrix+Theory&rft.pages=183&rft.pub=Dover&rft.date=1980&rft.isbn=9780486150277&rft.aulast=Eves&rft.aufirst=Howard+Whitley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DayVxeUNbZRAC%26pg%3DPA183&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988">. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, pp. 155, 162 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 162 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Knapp, Anthony W. (2006), Basic Algebra, <a href="/wiki/Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>, p. 121. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFröhlich,_A.Shepherson,_J._C.1955" class="citation cs2">Fröhlich, A.; Shepherson, J. C. (1955), "On the factorisation of polynomials in a finite number of steps", Mathematische Zeitschrift, 62 (1): 331–334, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01180640">10.1007/BF01180640</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119955899">119955899</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=On+the+factorisation+of+polynomials+in+a+finite+number+of+steps&rft.volume=62&rft.issue=1&rft.pages=331-334&rft.date=1955&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119955899%23id-name%3DS2CID&rft.issn=0025-5874&rft_id=info%3Adoi%2F10.1007%2FBF01180640&rft.au=Fr%C3%B6hlich%2C+A.&rft.au=Shepherson%2C+J.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall1969" class="citation book cs1">Hall, F. M. (1969). "Section 3.6". <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoab0000hall_v1">An Introduction to Abstract Algebra</a>. Vol. 2. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521084849" title="Special:BookSources/0521084849"><bdi>0521084849</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+3.6&rft.btitle=An+Introduction+to+Abstract+Algebra&rft.pub=Cambridge+University+Press&rft.date=1969&rft.isbn=0521084849&rft.aulast=Hall&rft.aufirst=F.+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoab0000hall_v1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerstein1975" class="citation book cs1">Herstein, I. N. (1975). "Section 3.9". <a rel="nofollow" class="external text" href="https://archive.org/details/topicsinalgebra00hers">Topics in Algebra</a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0471010901" title="Special:BookSources/0471010901"><bdi>0471010901</bdi></a>. <q>polynomial ring.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+3.9&rft.btitle=Topics+in+Algebra&rft.pub=Wiley&rft.date=1975&rft.isbn=0471010901&rft.aulast=Herstein&rft.aufirst=I.+N.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftopicsinalgebra00hers&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLam2001" class="citation cs2"><a href="/wiki/Tsit_Yuen_Lam" title="Tsit Yuen Lam">Lam, Tsit-Yuen</a> (2001), A First Course in Noncommutative Rings, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95325-0" title="Special:BookSources/978-0-387-95325-0"><bdi>978-0-387-95325-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Noncommutative+Rings&rft.pub=Springer-Verlag&rft.date=2001&rft.isbn=978-0-387-95325-0&rft.aulast=Lam&rft.aufirst=Tsit-Yuen&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2002" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002), Algebra, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95385-4" title="Special:BookSources/978-0-387-95385-4"><bdi>978-0-387-95385-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=Revised+third&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-0-387-95385-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1878556%23id-name%3DMR&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsborne2000" class="citation cs2">Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics, vol. 196, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-1278-2">10.1007/978-1-4612-1278-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98934-1" title="Special:BookSources/978-0-387-98934-1"><bdi>978-0-387-98934-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1757274">1757274</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+homological+algebra&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=2000&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1757274%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-1278-2&rft.isbn=978-0-387-98934-1&rft.aulast=Osborne&rft.aufirst=M.+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolynomial+ring" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1455652#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1455652#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <a href="https://www.wikidata.org/wiki/Q1455652#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/1070714/">FAST</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4175268-5">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85104701">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12270236s">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb12270236s">BnF data</a></li><li><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007563144605171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://www.idref.fr/031499066">IdRef</a></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Polynomial_ring&oldid=1254302004">https://en.wikipedia.org/w/index.php?title=Polynomial_ring&oldid=1254302004</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Commutative_algebra" title="Category:Commutative algebra">Commutative algebra</a></li><li><a href="/wiki/Category:Invariant_theory" title="Category:Invariant theory">Invariant theory</a></li><li><a href="/wiki/Category:Ring_theory" title="Category:Ring theory">Ring theory</a></li><li><a href="/wiki/Category:Polynomials" title="Category:Polynomials">Polynomials</a></li><li><a href="/wiki/Category:Free_algebraic_structures" title="Category:Free algebraic structures">Free algebraic structures</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_cleanup_from_June_2023" title="Category:Articles needing cleanup from June 2023">Articles needing cleanup from June 2023</a></li><li><a href="/wiki/Category:All_pages_needing_cleanup" title="Category:All pages needing cleanup">All pages needing cleanup</a></li><li><a href="/wiki/Category:Cleanup_tagged_articles_with_a_reason_field_from_June_2023" title="Category:Cleanup tagged articles with a reason field from June 2023">Cleanup tagged articles with a reason field from June 2023</a></li><li><a href="/wiki/Category:Wikipedia_pages_needing_cleanup_from_June_2023" title="Category:Wikipedia pages needing cleanup from June 2023">Wikipedia pages needing cleanup from June 2023</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_January_2021" title="Category:Articles needing additional references from January 2021">Articles needing additional references from January 2021</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_to_be_expanded_from_April_2022" title="Category:Articles to be expanded from April 2022">Articles to be expanded from April 2022</a></li><li><a href="/wiki/Category:All_articles_to_be_expanded" title="Category:All articles to be expanded">All articles to be expanded</a></li><li><a href="/wiki/Category:Articles_with_empty_sections_from_April_2022" title="Category:Articles with empty sections from April 2022">Articles with empty sections from April 2022</a></li><li><a href="/wiki/Category:All_articles_with_empty_sections" title="Category:All articles with empty sections">All articles with empty sections</a></li><li><a href="/wiki/Category:Articles_to_be_expanded_from_June_2020" title="Category:Articles to be expanded from June 2020">Articles to be expanded from June 2020</a></li><li><a href="/wiki/Category:Harv_and_Sfn_no-target_errors" title="Category:Harv and Sfn no-target errors">Harv and Sfn no-target errors</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"> This page was last edited on 30 October 2024, at 10:33 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Polynomial_ring&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-m4ftb","wgBackendResponseTime":153,"wgPageParseReport":{"limitreport":{"cputime":"0.876","walltime":"1.207","ppvisitednodes":{"value":11406,"limit":1000000},"postexpandincludesize":{"value":108662,"limit":2097152},"templateargumentsize":{"value":15107,"limit":2097152},"expansiondepth":{"value":15,"limit":100},"expensivefunctioncount":{"value":25,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":71156,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 857.812 1 -total"," 18.47% 158.438 1 Template:Reflist"," 17.14% 147.019 158 Template:Math"," 15.94% 136.747 8 Template:Citation"," 10.78% 92.507 1 Template:Ring_theory_sidebar"," 10.54% 90.440 1 Template:Sidebar_with_collapsible_lists"," 10.28% 88.221 1 Template:Short_description"," 10.22% 87.700 162 Template:Main_other"," 10.02% 85.963 1 Template:Cleanup"," 8.05% 69.026 1 Template:Authority_control"]},"scribunto":{"limitreport-timeusage":{"value":"0.441","limit":"10.000"},"limitreport-memusage":{"value":10681446,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAntonBivensDavis2012\"] = 1,\n [\"CITEREFEves1980\"] = 1,\n [\"CITEREFFröhlich,_A.Shepherson,_J._C.1955\"] = 1,\n [\"CITEREFHall1969\"] = 1,\n [\"CITEREFHerstein1975\"] = 1,\n [\"CITEREFLam2001\"] = 1,\n [\"CITEREFOsborne2000\"] = 1,\n [\"CITEREFSendraWinklerPérez-Diaz2007\"] = 1,\n [\"free_commutative_algebra\"] = 1,\n [\"free_commutative_ring\"] = 1,\n [\"minimal_polynomial\"] = 1,\n [\"multivariable\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Abs\"] = 1,\n [\"Anchor\"] = 3,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 7,\n [\"Cite book\"] = 2,\n [\"Cite web\"] = 2,\n [\"Cleanup\"] = 1,\n [\"Empty section\"] = 1,\n [\"Expand section\"] = 1,\n [\"Further\"] = 1,\n [\"Harv\"] = 4,\n [\"Harvnb\"] = 6,\n [\"I sup\"] = 7,\n [\"Lang Algebra\"] = 1,\n [\"Main\"] = 14,\n [\"Math\"] = 158,\n [\"Mvar\"] = 172,\n [\"Nowrap\"] = 6,\n [\"Reflist\"] = 1,\n [\"Ring theory sidebar\"] = 1,\n [\"See also\"] = 1,\n [\"Short description\"] = 1,\n [\"Sub\"] = 5,\n [\"Sup\"] = 2,\n [\"Unreferenced section\"] = 1,\n [\"Val\"] = 2,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-bb444","timestamp":"20241125133618","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Polynomial ring","url":"https:\/\/en.wikipedia.org\/wiki\/Polynomial_ring","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1455652","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1455652","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-11-20T08:07:01Z","dateModified":"2024-10-30T10:33:28Z","headline":"algebraic structure"}</script> </body> </html>

CINXE.COM

Polynomial ring - Wikipedia