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<h3> a  <varclass""F<var

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fieldvar="Arg><var  a subfield of acommutative ring < class="Argdivclass=ContSect

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<h4>58.1 <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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<h5< classpAemivisionemin="GAPstrong     < "<  multiplicative . strongkg everyring vector  divisionring overitself Note beingdivisionisthus a propertya get because isusually represented a  spacep>

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<p>The

<p><a id="X7A5AE30E7C0F457C" name="X7A5AE30E7C0F457C"></span

<h5>58.1java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<divclass>table="" widthjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>A <em>ieldjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div classGAPpromptgap&;/an <
< class=""gapgt<spantrue
true
< span=GAPpromptgap&;/pan =":QuaternionAlgebra(Rationals;;    division<spanjava.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
true
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false
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false
<rep> e classfuncField/code  the smallest  < class=SimpleMath"K</hatcontains the elements <pan class"SimpleMath"var =Arg"<> .span orthe  <spanclass"K<span that containscontains  elementsin  list var=rg></> If no subfield<varclass"ArgF  givenspan="SimpleMath><an>isconstructedasa over ,i. the left domainof span=SimpleMath>K/>isclass="impleMath/panCalledafieldvar"Arg>F</   var=field  var=rg>> elementsvar=Arg>/>as   over< =<var./>

<p><a id="X871AA7D58263E9AC" name="X871AA7D58263E9AC"></a></p>

<h5>58.1-3 Field</h5>

<div class
< classfunc< class"" width="00%">trclass="dleft><odeclass"func>#8227; Field</code( [<#X7F2CAA9E7A16913D"><span ="RefLink">81-1<>/)
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<p><a id="X7D9F7FD4786691EE" name="X7D9F7FD4786691EE"></a></p>

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< classfunc> classdiv="">tableclasswidth100">td class=""><code =unc">;DefaultField/> arclass>ist>)< class"&;functionnbsp)td>/>table<div
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CF(9)
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<panspanclass"GAPprompt>gapt/span><spanclass="">f1 (EB(   ::=efaultField 5));</>
NFNF5 NF
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falsejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
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<h5>58.1-5 DefaultFieldByGenerators</h5

div=func class"width"00"tr<tdclass=tdleft> class=">#;DefaultFieldByGenerators([varArg/varArg./>/< =tdright;operation<>/
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<<<>586 </5>

<div class="func"><table class
<p>generatorsgenerators

<1 <java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

<h5>58.1-7 GeneratorsOfField</h5>

<divclass=">table class"" ="100%>><td=<h558- DivisionRingByGenerators/java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 40
p>with , andinversesattributeisthesame< =><<code<hrefhap58tml"<panclassRefLink"58.-/span<>)</

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<h5>58.2-6 DegreeOverPrimeField

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<h5.-8 RootOfDefiningPolynomial</h5>

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<p><a id="X83490C65819D85FE" name="X83490C65819D85FE

<h5>58> norm trace <span=SimpleMathα/panare span"SimpleMath"K/,andspan=SimpleMathμαspan and class">_αspan arepolynomialsover <panclass"""><span,< class"χ<  a powerof< class=SimpleMath>μα<span, andthedegree of spanclass"SimpleMath>_α/> equals<ivclass"func>< class"func" width="00%tr classtdleft< classfunc&; RootOfDefiningPolynomial<code(var=F<ar)td< ="right(nbspattributebsp)</>r>tablediv>

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<p><a id="X8738C6687D784BB5>< ="X8738C6687D784BB5 nameX8738C6687D784BB5"><a>/>

<5583-MinimalPolynomial<h5>

<div class"><ableclass=unc""widthwidth=100%">tr<tdclassdleft><code=&8227;MinimalPolynomial<code(arclassArgF/,<classArg><var  class>indvar>/d<dclass(nbspoperationnbsp)<d</>table<>>
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</pre></div>

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[ -E(5), -(5)^, -E(5)3, -E(5)^4 ]
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