CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> field in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> field </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8176/#Item_17" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Fields</title></head> <body> <blockquote> <p>This entry is about the notion in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>. For the different concept of the same name in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> see at <em><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a></em>, and for that in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> see at <em><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></em>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> </div> </div> <h1 id="fields">Fields</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#constructive'>Constructive notions</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#category'>Category of fields</a></li> <li><a href='#AccSketch'>Accessibility and sketchability</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A field is a mathematical structure with the usual arithmetic operations of <a class="existingWikiWord" href="/nlab/show/addition">addition</a>, <a class="existingWikiWord" href="/nlab/show/subtraction">subtraction</a>, <a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a>, and <a class="existingWikiWord" href="/nlab/show/division">division</a> of non-zero elements.</p> <p>The original concept of a field was developed by <a class="existingWikiWord" href="/nlab/show/Gustav+Lejeune+Dirichlet">Gustav Lejeune Dirichlet</a> and <a class="existingWikiWord" href="/nlab/show/Richard+Dedekind">Richard Dedekind</a> in 1871 under the name “body” (Körper in German), and referred solely to <a class="existingWikiWord" href="/nlab/show/subfields">subfields</a> of the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> (cf. <a href="#DirichletDedekind71">Dirichlet & Dedekind 1871</a>). Later in 1893 <a class="existingWikiWord" href="/nlab/show/Heinrich+Weber">Heinrich Weber</a> generalized the notion of field to today’s definition of field as an arbitrary <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative</a> <a class="existingWikiWord" href="/nlab/show/division+ring">division ring</a> (cf. <a href="#Weber93">Weber 1893</a>). The word “field” itself in the English language was coined by <a class="existingWikiWord" href="/nlab/show/E.+Hastings+Moore">E. Hastings Moore</a> in 1893 (cf. <a href="#Moore93">Moore 1893</a>) to refer to the algebraic structure.</p> <h2 id="definitions">Definitions</h2> <p>Classically:</p> <div class="num_defn" id="classical"> <h6 id="definition">Definition</h6> <p>A <strong>field</strong> is a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> in which every nonzero element has a multiplicative <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \neq 1</annotation></semantics></math> (which may be combined as: an element is invertible if and only if it is nonzero).</p> </div> <p>Fields are studied in <em>field theory</em>, which is a branch of <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a>.</p> <p>If we omit the commutativity axiom, then the result is a <a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a> or <a class="existingWikiWord" href="/nlab/show/division+ring">division ring</a> (also in some contexts simply called a “field”). For example, the <a class="existingWikiWord" href="/nlab/show/free+field+%28algebra%29">free field</a> of Cohn and Amitsur is in fact noncommutative.</p> <h3 id="constructive">Constructive notions</h3> <p>Fields are (arguably) not a purely <a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> notion in that they don't form an <a class="existingWikiWord" href="/nlab/show/algebraic+category">algebraic category</a> (see <a href="#category">discussion below</a>). For this reason, it should be unsurprising that in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> (including the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>) there are different inequivalent ways to define a field. In this case the classical definition is not usually the best one; for instance, the real numbers do not satisfy it. There are several potential replacements with their own advantages and disadvantages. These include:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+field">discrete field</a></li> <li><a class="existingWikiWord" href="/nlab/show/Heyting+field">Heyting field</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+Heyting+field">weak Heyting field</a></li> <li><a class="existingWikiWord" href="/nlab/show/Kock+field">Kock field</a></li> <li><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+local+ring">weak local ring</a></li> </ul> <p> <div class='num_defn' id='discrete'> <h6>Definition</h6> <p>If we replace “an element is invertible iff it is nonzero” in Definition <a class="maruku-ref" href="#classical"></a> by “an element is invertible <a class="existingWikiWord" href="/nlab/show/xor">xor</a> it equals zero” (which is equivalent in <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a> but stronger in <a class="existingWikiWord" href="/nlab/show/constructive+logic">constructive logic</a>), then we obtain the notion of <strong>discrete field</strong>. This condition means that every element is either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> or invertible, and it also implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\neq 1</annotation></semantics></math>.</p> </div> </p> <p>Such a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is ‘discrete’ in that it decomposes as a coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>⊔</mo><msup><mi>F</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">F = \{0\} \sqcup F^\times</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">F^\times</annotation></semantics></math> is the subset of invertible elements). An advantage is that this is a <a class="existingWikiWord" href="/nlab/show/coherent+logic">coherent theory</a> and hence also a <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a>; for this reason <a href="#Johnstone77">Johnstone</a> calls such fields <strong>geometric fields</strong>. A disadvantage is that this axiom is not satisfied (constructively) by the ring of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> (however these are defined), although it is satisfied by the ring of <a class="existingWikiWord" href="/nlab/show/rational+number">rational</a> (or even <a class="existingWikiWord" href="/nlab/show/algebraic+number">algebraic</a>) numbers and by the <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a>s as usual.</p> <p> <div class='num_defn' id='heyting'> <h6>Definition</h6> <p>If we interpret ‘nonzero’ in Definition <a class="maruku-ref" href="#classical"></a> as a reference to a <a class="existingWikiWord" href="/nlab/show/tight+apartness+relation">tight apartness relation</a>, thus defining the apartness relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>#</mo></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x # y</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x - y</annotation></semantics></math> is invertible, then we obtain the notion of <strong><a class="existingWikiWord" href="/nlab/show/Heyting+field">Heyting field</a></strong>. (As shown <a href="/nlab/show/local+ring#internal">here</a>, the ring operations become strongly extensional functions.) In addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>#</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\# 1</annotation></semantics></math>, the condition then means that every element apart from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> is invertible.</p> </div> </p> <p>This is how ‘practising’ constructive analysts of the Bishop school usually define the simple word ‘field’. An advantage is that the (located Dedekind) <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> form a Heyting field, although (for example) the (less located) <a class="existingWikiWord" href="/nlab/show/MacNeille+real+number">MacNeille real number</a>s need not form a Heyting field; another disadvantage is that this is not a coherent axiom and so cannot be <a class="existingWikiWord" href="/nlab/show/internalization">internalized</a> in as many categories.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>If we replace “an element is invertible iff it is nonzero” in Definition <a class="maruku-ref" href="#classical"></a> by “an element is noninvertible iff it is zero” (which is equivalent in <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a> but incomparable in <a class="existingWikiWord" href="/nlab/show/constructive+logic">constructive logic</a>), we obtain the notion of <strong><a class="existingWikiWord" href="/nlab/show/weak+Heyting+field">weak Heyting field</a></strong> (cf. <a href="#Richman20">Richman (2020)</a>) or <strong><a class="existingWikiWord" href="/nlab/show/Johnstone+residue+field">Johnstone residue field</a></strong> (cf. <a href="#Johnstone77">Johnstone (1977)</a>), which is not quite the same as the <a class="existingWikiWord" href="/nlab/show/residue+fields">residue fields</a> of <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>). In addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\neq 1</annotation></semantics></math>, this condition means that every noninvertible element (i.e. element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>y</mi><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x y\neq 1</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>) is zero.</p> </div> </p> <p>An advantage is that even more versions of the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> (including the <a class="existingWikiWord" href="/nlab/show/MacNeille+real+number">MacNeille real number</a>s) form a weak Heyting field; disadvantages are that this axiom is not coherent either and that a residue field lacks an <a class="existingWikiWord" href="/nlab/show/apartness+relation">apartness relation</a> (in particular, the MacNeille reals have no apartness). Another example of a weak Heyting field include the set of <a class="existingWikiWord" href="/nlab/show/Laurent+series">Laurent series</a> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.</p> <p>Every discrete field is also a Heyting field, and every Heyting field is also a weak Heyting field. However, a Heyting field is a discrete field if and only if its <a class="existingWikiWord" href="/nlab/show/apartness+relation">apartness relation</a> is a <a class="existingWikiWord" href="/nlab/show/decidable+relation">decidable relation</a>.</p> <p>A weak Heyting field is a Heyting field if and only if it is a <a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>. Furthermore, the quotient ring (or ‘residue ring’) of any local ring by its ideal of noninvertible elements is a Heyting field; in particular, it is a <a class="existingWikiWord" href="/nlab/show/residue+field">residue field</a>. On the other hand, not every weak Heyting field is even a local ring (the MacNeille reals are not), so not every residue field is the residue ring of any local ring. The name “residue field” as defined in <a href="#Johnstone77">Johnstone (1977)</a> comes from the fact that these fields are precisely the residue rings of <em>weak local rings</em> (rings in which the noninvertible elements form an ideal).</p> <p>Counterexamples were remarked above, but to be explicit: The (located Dedekind) <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> form a Heyting field which need not be discrete. The <a class="existingWikiWord" href="/nlab/show/MacNeille+real+number">MacNeille real number</a>s form a weak Heyting field which need not be Heyting; see section D4.7 of <em><a class="existingWikiWord" href="/nlab/show/Elephant">Sketches of an Elephant</a></em>.</p> <p>The three definitions above do not exhaust the possible constructive notions of field. For instance, in <a href="#MRR87">MRR87</a> the unadorned word <strong>field</strong> is defined like a Heyting field above, but with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>#</mo></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math> being an arbitrary <a class="existingWikiWord" href="/nlab/show/inequality+relation">inequality relation</a> rather than a tight apartness. If the inequality is the <a class="existingWikiWord" href="/nlab/show/denial+inequality">denial inequality</a>, this reproduces the original classical definition, which in <a href="#Johnstone77">Johnstone77</a> is called a <strong>field of fractions</strong> since they are precisely the fields of fractions of “weak <a class="existingWikiWord" href="/nlab/show/integral+domains">integral domains</a>” (defined as rings in which the product of two nonzero elements is nonzero). In <a href="#MRR87">MRR87</a> a <strong>denial field</strong> is defined to be a Heyting field with respect to the <a class="existingWikiWord" href="/nlab/show/denial+inequality">denial inequality</a> in which additionally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>.</p> <p> <div class='num_remark' id='TrivialRingAsAField'> <h6>Remark</h6> <p>In <a href="#LombardiQuitté2010">LombardiQuitté2010</a>, the authors’ definitions of discrete field and Heyting field do not include the non-equational axiom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \neq 0</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>#</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \# 0</annotation></semantics></math> respectively. With such a definition, the <a class="existingWikiWord" href="/nlab/show/trivial+ring">trivial ring</a> counts as a discrete field as well as a Heyting field and constitutes the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in the <a class="existingWikiWord" href="/nlab/show/category">categories</a> of such fields.</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="category">Category of fields</h3> <p>Fields are not as well-behaved <a class="existingWikiWord" href="/nlab/show/category+theory">categorically</a> as most other common algebraic structures (<a class="existingWikiWord" href="/nlab/show/groups">groups</a>, <a class="existingWikiWord" href="/nlab/show/rings">rings</a>, <a class="existingWikiWord" href="/nlab/show/modules">modules</a>, etc.). In particular, <a class="existingWikiWord" href="/nlab/show/Field">Field</a>, the <a class="existingWikiWord" href="/nlab/show/category">category</a> of fields and field <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> (a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the category <em><a class="existingWikiWord" href="/nlab/show/Rings">Rings</a></em> of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> and <a class="existingWikiWord" href="/nlab/show/ring+homomorphisms">ring homomorphisms</a>) is not <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> or <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a>, although it is <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible</a>.</p> <p>In particular, it lacks a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> and also lacks an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> (though it has a <a class="existingWikiWord" href="/nlab/show/weakly+initial+set">weakly initial set</a>, namely the set of <a class="existingWikiWord" href="/nlab/show/prime+fields">prime fields</a>, hence has a “<a href="multi-adjoint#MultiInitialObjectsInFields">multi-initial object</a>”). In particular, it is therefore not <a class="existingWikiWord" href="/nlab/show/algebraic+category">algebraic</a> or <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a>.</p> <h3 id="AccSketch">Accessibility and sketchability</h3> <p><a class="existingWikiWord" href="/nlab/show/Field">Field</a> is <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible</a>, even <em>finitely</em> accessible, and therefore can be presented as the category of models (in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) of a mixed limit-colimit <a class="existingWikiWord" href="/nlab/show/sketch">sketch</a>. It is moreover straightforward to write down such a sketch.</p> <p>We suppose as given to start with a <a class="existingWikiWord" href="/nlab/show/limit+sketch">limit sketch</a> whose models are <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> denoting the ring. We can construct via limit constructions a subobject <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>↪</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">I\hookrightarrow F</annotation></semantics></math> consisting of the invertible elements, as the equalizer of the two maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>×</mo><mi>F</mi><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>F</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> F \times F \;\rightrightarrows\; F,</annotation></semantics></math></div> <p>the first being given by multiplication and the second by the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>×</mo><mi>F</mi><mo>→</mo><mo>*</mo><mover><mo>→</mo><mn>1</mn></mover><mi>F</mi></mrow><annotation encoding="application/x-tex">F\times F \to * \overset{1}{\to} F</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> is terminal and the map labeled “1” picks out the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">1\in F</annotation></semantics></math>. We now assert that if we take the pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mn>0</mn></msup></mtd></mtr> <mtr><mtd><mi>I</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>F</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{P & \overset{}{\to} & * \\ \downarrow && \downarrow^0\\ I& \hookrightarrow & F,} </annotation></semantics></math></div> <p>where the map labeled “0” picks out the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">0\in F</annotation></semantics></math>, then the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is initial (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> is not invertible, or equivalently not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>), and moreover the pullback is also a pushout (i.e. every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> or invertible). Of course, in making these last two assertions we use the fact that we are allowing ourselves a limit-colimit sketch instead of just a limit sketch.</p> <p>Note that this gives us the notion of <em>discrete</em> field (see the <a href="#constructive">constructive definitions</a> above). The other constructive notions of field can also be described as models for different limit-colimit sketches.</p> <h2 id="Examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <p>There are the fields of:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>,</p> </li> <li> <p>(all) <a class="existingWikiWord" href="/nlab/show/algebraic+numbers">algebraic numbers</a>, (Kontsevich-Zagier) <a class="existingWikiWord" href="/nlab/show/period">period</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a>).</p> </li> </ul> <p>Other examples</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>F</mi></mstyle> <mrow><msup><mi>p</mi> <mi>n</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{F}_{p^n}</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a prime</p> </li> <li> <p>algebraic <a class="existingWikiWord" href="/nlab/show/number+field">number field</a>s – finite degree extensions of the field of rational numbers</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+field">function field</a>s</p> </li> </ul> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>The canonical <a class="existingWikiWord" href="/nlab/show/ring+object">local ring object</a> of the <a class="existingWikiWord" href="/nlab/show/Zariski+site">gros Zariski topos</a> of any <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> (given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>↦</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \mapsto \Gamma(S, \mathcal{O}_S)</annotation></semantics></math>, that is to say, the affine line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mi>S</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^{1}_{S}</annotation></semantics></math>) is in fact moreover a field object, where the latter is defined by requiring that Definition <a class="maruku-ref" href="#classical"></a> holds in the internal logic of this topos. For a proof, see Proposition 2.2 in the article <a href="http://www.sciencedirect.com/science/article/pii/0022404976900025">Universal projective geometry via topos theory</a> of Anders Kock. The ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a>) in the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">sheaf topos</a> (i.e. the petit Zariski topos) is a <a class="existingWikiWord" href="/nlab/show/residue+field">residue field</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/reduced+scheme">reduced</a> scheme.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prefield+ring">prefield ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reciprocal+ring">reciprocal ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/division+ring">division ring</a>, <a class="existingWikiWord" href="/nlab/show/division+monoid">division monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subfield">subfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a>, <a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/possibly+trivial+field">possibly trivial field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/residue+field">residue field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+field">global field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/number+field">number field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+field">function field</a> (over a finite field)</p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+field">topological field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+field">graded field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+theory+of+fields">model theory of fields</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinity-field">infinity-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock+field">Kock field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+setoid">field setoid</a></p> </li> </ul> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></th><th><a class="existingWikiWord" href="/nlab/show/reduced+ring">reduced ring</a></th><th><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/reduced+local+ring">reduced local ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+integral+domain">local integral domain</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semisimple+ring">semisimple ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field">field</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Weil+ring">Weil ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field">field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field">field</a></td></tr> </tbody></table> <h2 id="references">References</h2> <ul> <li id="DirichletDedekind71"> <p><a class="existingWikiWord" href="/nlab/show/Gustav+Lejeune+Dirichlet">Gustav Lejeune Dirichlet</a> (1871), <a class="existingWikiWord" href="/nlab/show/Richard+Dedekind">Richard Dedekind</a> (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn (<a href="https://books.google.com/books?id=SRJTAAAAcAAJ&pg=PA424">Google Books</a>)</p> </li> <li id="Weber93"> <p><a class="existingWikiWord" href="/nlab/show/Heinrich+Weber">Heinrich Weber</a> (1893), “Die allgemeinen Grundlagen der Galois’schen Gleichungstheorie”, Mathematische Annalen (in German), 43 (4): 521–549, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007%2FBF01446451">doi:10.1007/BF01446451</a>, <a href="https://www.worldcat.org/issn/0025-5831">ISSN:0025-5831</a>, <a href="https://zbmath.org/?format=complete&q=an:25.0137.01">JFM:25.0137.01</a>, <a href="https://api.semanticscholar.org/CorpusID:120528969">S2CID:120528969</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Moore93"> <p><a class="existingWikiWord" href="/nlab/show/E.+Hastings+Moore">E. Hastings Moore</a> (1893), “A doubly-infinite system of simple groups”, Bulletin of the American Mathematical Society, 3 (3): 73–78, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1090%2FS0002-9904-1893-00178-X">doi:10.1090/S0002-9904-1893-00178-X</a>, <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1557275">MR:1557275</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="MRR87"> <p>Ray Mines, <a class="existingWikiWord" href="/nlab/show/Fred+Richman">Fred Richman</a>, Wim Ruitenburg, <em>A course in constructive algebra</em>, Universitext, Springer, 1987.</p> </li> <li id="Elephant"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a>, Part D. The <a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a> for fields is discussed in section D3.1.11(b).</p> </li> <li id="Johnstone77"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Rings, Fields, and Spectra</em>, Journal of Algebra <strong>49</strong> (1977) 238-260 <a href="https://doi.org/10.1016/0021-8693%2877%2990284-8">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Olivia+Caramello">Olivia Caramello</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>De Morgan’s law and the theory of fields</em> (<a href="http://arxiv.org/abs/0808.1972">arXiv:0808.1972</a>)</p> </li> <li id="LombardiQuitté2010"> <p><a class="existingWikiWord" href="/nlab/show/Henri+Lombardi">Henri Lombardi</a>, <a class="existingWikiWord" href="/nlab/show/Claude+Quitt%C3%A9">Claude Quitté</a> (2010): <em>Commutative algebra: Constructive methods (Finite projective modules)</em> Translated by Tania K. Roblo, Springer (2015) (<a href="https://link.springer.com/book/10.1007/978-94-017-9944-7">doi:10.1007/978-94-017-9944-7</a>, <a href="http://hlombardi.free.fr/CACM.pdf">pdf</a>)</p> </li> <li id="Richman20"> <p><a class="existingWikiWord" href="/nlab/show/Fred+Richman">Fred Richman</a>, <em>Laurent series over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></em>. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [<a href="https://doi.org/10.1080/00927872.2019.1710166">doi:10.1080/00927872.2019.1710166</a>]</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marc+Bezem">Marc Bezem</a>, <a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Cagne">Pierre Cagne</a>, <a class="existingWikiWord" href="/nlab/show/Bj%C3%B8rn+Ian+Dundas">Bjørn Ian Dundas</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+R.+Grayson">Daniel R. Grayson</a>: Chapter 8 of: <em><a class="existingWikiWord" href="/nlab/show/Symmetry">Symmetry</a></em> (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://unimath.github.io/SymmetryBook/book.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 19, 2024 at 14:53:14. See the <a href="/nlab/history/field" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/field" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8176/#Item_17">Discuss</a><span class="backintime"><a href="/nlab/revision/field/74" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/field" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/field" accesskey="S" class="navlink" id="history" rel="nofollow">History (74 revisions)</a> <a href="/nlab/show/field/cite" style="color: black">Cite</a> <a href="/nlab/print/field" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/field" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>