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</div=">A>ivision ring</> in<classpkg<strongis nontrivialassociative algebra "Argaramultiplicativeinverse each . < =GAP     space  possiblyitself     ring not    can   represented space
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<p>A <em>division ring</em> is apue

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<p>generators with respect to addition, multiplication, and taking class

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< class"tablefunc =00%tr>td="code="8227RootOfDefiningPolynomialcode< ""F>>> "right;&;)d><<>
<p>is a root in the field <var class="Arg">F</var> of its defining polynomial as a field extension over the left acting domain of <var class="Arg">F</var>. The defining polynomial can be computed

<p><a id="X82718B3B818DC699" namepis set

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<h5>58.3-1 GaloisGroup</h5>

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<ple<pre
<span class="GAPprompt<spanpreturnsthepolynomial that the  span =impleMath"X-span< ="<spanruns  of < class"Arg<ar  field classArgz<var inthe extension<varclass"varover var class">/> The isreturned univariate overvar classArg<varin theindeterminate number < class>inum/ defaulting to 1/>
x_1^4+<ivclass"func><table ="func" width="100x_1^2+(-E(4)> polynomialissometimesalsocalledthe <em>characteristic<m of>thepolynomialthatthe of< =""X -)</>wherespanclass"<pan  over  conjugates ofof<ar univariate over<class"Arg<>intheindeterminatenumbervar"><var> defaulting1</>>
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<p><a id="X80FE7E017C2D255C" name

<h5>58.3-3<><div>

<divclassThis polynomial  also theemcharacteristicpolynomial><  classArgvar.field <class>var"L<> &; var=rg>K<var>/.Therefore areinstalledforcodeclass>CharacteristicPolynomial<code<href".tml87FA0A727CDB060B> class>.1/>)  code classuncTracePolynomialthe ofextensions/java.lang.StringIndexOutOfBoundsException: Index 485 out of bounds for length 484
<p>returns the polynomial that is the 

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<div class="example"><pre>
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<p><a id="X838515278587FF01" name<><>58.="><ode traceof   "z> fields"><  <varclass"Arg><>aregiven  trace  w..e  extension < =""var"L<ar> ;< =">/><span only field classArg">var> is giventhen <odeclass"code>(  class<ar>/>istaken defaultthesubfield<classArgK<varandiffieldisgiven <codeclass"DefaultField << classrg"zvar )/codeistakenforvarclassArgL>/p

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<h558.35 <panclass"">Traces

<div"func>race/> trace ofthethefieldelement<varclass"Arg</>If fieldsclassArgL/ <class"</aregiventhetrace iscomputed w..fieldextension <impleMathvarArg<>gt<var"/</if field">var given<classcodeeftActingDomain=Arg></>  defaultfor <classK>, iffieldgiven< classcodevarclass">varcode  default<class""<var<p>
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<h5>58.3-6 Conjugates</h5>

<divdiv class"func><tableclass<pan class="APpromptgapgt;</span span=GAPinputTrace())  (CF)CF class&;<>< class">( 8;  ( (E(8</an
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<p>a=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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