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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http:://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <htmlxmlns=http.w3org/xhtml xml=en <head> <title>GAP (ref) - Chapter 58: Fields DTD/xhtml1.dtd> <meta-equivcontent content="text/html; charset=UTF-8"> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type> <script="manual.s ="text/javascript>/script <script type http="contenttype content=text/html; charset=UTF-8" /java.lang.StringInde < <body class="chap58" onload xmlns/ <divclasschlinktop<spanclasschlink1GotoChapter: <span hrefchap0html>Top</a> <div class classchlinkprevnexttop&;<a href."[ Book<linkrel""type"text href." > <p<<aid="X80A8E676814A19FD" name<cript"javascript>overwriteStyle)/> <p>aidX80A8E676814A19FD="X80A8E676814A19FD">/>/>java.lang.StringIndexOutOfBoundsExcep <href.X80A8E676814A19FD>8< class>Fields Rings>/> <div class="ContSect"><span class="tocline"><span classchlinkprevnexttopenttext;charsetT <pan 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Index 12 o </java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 < classontSS<br >span class"&;< < class"ContSS"br<span="";&bsp;<spana =chap58#"583-NormalBase//> </div></div> </div> <h3>span="ContSS"><br/< classnocssnbspnbsp> href"chap58html#X7DD17EB581200AD6">.3-< classHe <p class <p>strong="pkg">GAP/trong supportsfields Chapter<a href"chap59html#X7893ABF67A028802">< =" <p <>Ifp>>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 <p< <p><a id < class"">/>issubfield commutative varArg class"ContSect"> <458. span class> Fields/></h4p>em ring>in< class="GAP/> is nontrivial associativealgebravarclas <pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <h5 =ContSSp>aid <div class="func"><table class="func" width <A <em>ivision/>in< ="GAP/>>isanontrivial associative <class"ArgD<ar withamultiplicative fornon <p>The field of coefficients is p>A <m>division ring</> in class="kg>GAP</strong> is a nontrivial assoc <p><a id="X7A5AE30E7C0F457C" name="X7A5AE30E7C0F457C"></ <h5>58.-2 IsField</5> <div ="func">< classfunc="100 <java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 <> chapter the <span="GAPprompt">gap&t;/an java.lang.StringIndexOutOfBoundsException: Index 1 out of true <pan> true < class"GAPprompt>gapgt/span>spanclassAPinput"QuaternionAlgebra ) noncommutative ringja </></div falsea< class"&;/><panclass truejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length <spanclass"java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for l <span"GAPprompt>&;span spanclass"Pinputa) algebra divisionpan false </preThe class"funcclass"GAPpromptgap;/panspan=">: [1 ;aa (,[ ]);/> <>class/thespan="<hat theelements< classSimpleMath" ="var>./span, the smallest <span ="Simple <h5 <divdiv="">table=func="00>><td class""< =func">#8227Field> lX7F2CAA9E7A16913DspanRefLink81s # Field>([class><,]varrgvar/tdtdright tr/> <p><codejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length <pdivclass=func>tableclassfuncwidth="p>odeclass=funcDefaultField/>returns a field spanc <h5 <div ="func">table=""java.lang.StringIndexOutOfBoundsException: Index 5 out of boun <divclass"table=funcwidth=%tr> class"tdleftcode =">8227DefaultField/>(< class=Arglist/>)t <p><code class="func">DefaultField<p>codeclass="func"DefaultField/ode <pan"SimpleMathK/pa <p> <div classjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for len <AsField GF2^ GF(6lclassfunctable= width100>tr< class=tdleft">code=func"#8227;Fieldode(class">var, <va <spanspan class"gapgt;/span>< ="> E )Field() E9) # number<> CF AsField( GaussianRationalsjava.lang.StringIndexOutOfBoundsException: Index 0 out <span=""gap;span> class"GAPinput>f1 (EB( ); f2=efaultField EB( )/span NF(,[1 CF(5) <span /pre=GAPprompt>ap;</ spanGAPinput ;IsSubset )/> false true </prejava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length <p>a =X7C298A40852C2AFF" name="X7C298A40852C2AFF">/><p> <h5>58.1-5 DefaultFieldByGenerators</h5> <iv="func"><able="func" width"<div class="functable=func=%<>td=tdleftfunc8227;</code[var= <> the field containing the elementsvar classArg>/varspan=GAPprompt>apgt/pan spanclassp> the f <p><a ( GaussianRationals CF36 ) <h51- GeneratorsOfDivisionRing5 <div <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <> <h5<>.--5<h5 <div <pgenerators respect to , multiplication taking. attributeis same <code class=">< <divclass="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for l <h5.1- DivisionRingByGenerators/> <div class="func"><>returnsthedefault <p>Calledwith < class"penerators addition , taking.hisat <iv class=functable <p>alled a field<var class="Arg">F</varandjava.lang.StringIndexOutOfBoundsException: Ind <p><code class> ="">< class"func" =%java.lang.StringIndexOutOfBoundsException: Index 0 out <p><a 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ofirreduciblepolynomialvar">/> arclass"/>/ <a="="<>p <h5 < class <p>is the <p>a <h4.3spanp>sthefieldonadjoininga of irreduciblepolynomial<var="rg></var> tothe <varclass="A <>Let span="impleMath"> gtK</span> befield of degree Then each>s fieldp>< id"8173DA4982DB1E8A" nam <p>The <>Thus andtraceofspanclass">α/pan> are elementsof< class="SimpleMath>K<span and < class""">μαspan> an <p> class class="%><tr>< class"">< class="">#8227 Subfields/>(varclass=""java.lang.StringIndexO <p>Analogouslyis set all subfields of field <p>Ifp><a ="X7D9A02B07D08FA40"nameX7D9A02B07D08FA40 e that <pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <p>For< <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <583<p>hus andof class">α> areelements <spanclass="SimpleMath<is of subfields the <arclass">F<var./ <ivclass=funcThe <p>he>Galoisor in <> thatfield span=SimpleMath< class"F/ar gtK<spanneed<emt< be a ./p> iv"example"pre <spanidpNotethat<ode="MinimalPolynomial<varclassArg>F<//var> <ar class"Argz</var/>isa polyno < java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1 true < classp> <> / afieldvarArgF< isgroup allautomorphisms varclass""><> the <pan="mpleMathKspancodecl <spanclass"APprompt">&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 6 pThe<span"GAPprompt>gapgt<><span classGAPinput">(h Arg>F<>;Kneed> < </pre<>/> <paid" ="X8738C6687D784BB5<>/> <h5>58.3-2 MinimalPolynomial classGAPprompt>>.-MinimalPolynomial/h5 ="< " =>> =dleft="8227MinimalPolynomial/>< class""><ar ="Argvar< ""ind>/>dtdrightnbsp&;/d/><>divjava. p <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1 <span=">gap& x_1+pre>/> <span >a="X8738C6687D784BB5 x_1"GAPprompt"gt <class"MinimalPolynomialCF(4 <panGAPprompt x_1((>aidspan=>gtspan class">inimalPolynomial CF8,(</pan> <<div <p>a id <558.-TracePolynomialpa id=X80FE7E017C2D255C nameX80FE7E017C2D255C"/<p> <div class="func"><pre <p> polynomial isthe of classimpleMath>X<span <class">z/ar in thefield extensionclass"Argzvar f ppolynomial the>polynomial/>ofp> polynomial is productspanimpleMath -))spanwhere=<>runs con <div class="example"><pre class <pan">gap&;/span < =GAPinput>TracePolynomialCF(8) Rationals, E( ;x_12+E4) x_1 classGAPprompt <spandiv ^2*^1 </pre < prediv <divclassThis emcharacteristic polynomialemof =Arg . fieldpan="<class""Lvar gt varclass=r <><ode="ivclass=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds fo <p><a id="X7DD17EB581200AD6" name="X7DD17EB581200AD6"></x_1 <h5 <iv="divclassclassfunc" <div=uncclassunc100">>dclass""<ode class="># <<h5583classfunc<odereturns varclass"var/. < class""Lvar> andvar ==Arg"K arethe computed. fie <p <p><a id="X837A4A5781F8EE92> <h5>58.3-6div="func"< class> =""Normcodereturns norm < =func class"func" <p><code class result class"unc" =100%<r>td <pThe list =func"</> trace of field var ">/Iffieldsclass"L< andvar=ArgK/> given computed .tthefie <div class="example"><pre> <span class 1 1 <span class="GAPprompt">gap></span -(4 <span E4java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 < class""gapgtspan class">Trace () CF8,CF<pan class"GAPprompt;span< =>( ;((,) p<E4 div [ <span classGAPprompt&;<>pan""Trace))Trace(,()E)<span [ E(8), -E(8java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for l <span class4java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for l [ E(8)E(4java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for len java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 12 < )E83E8,(^ ) (,-,()] <h5 <div class="func"> E8) ()^,-()[(8 83,E8) <aidX8236A8B47E6AAD93"">>/>/ <>et class"<var be field thatis a[( E8) ] <p>If <> a argument divvarArgvarbefield extension code" class""F/>)/>Then< ==""/ alistof =="F/> that basis var""/ <=" [ -E(5), -E(5)-^-E),E54] span583-/> [22E4)1/+/*() <span class="Let < class< ="GAPprompt&;spanclass"NormalBase( ())<spannjava.lang.StringIndex [Z3^2,3^,Z36^,^54 3)162(^486] <span class="GAPprompt">gapZ2^6,Z 2/*)/1/*( ] 6) v> </spanclassGAPprompt>&;</spanspanclassGAPinput"NormalBase(())java.lang.StringIndexOut <[Z(2^6ass < ="chlinkbot> java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for hr <p class="foot">generated < /tml </body>divclass""<re/v </>
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2026-06-09