Quelle chap58.html

Sprache: HTML

products/Sources/formale Sprachen/GAP/doc/ref/chap58.html

?version=-"

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.script src="manualjs/javascript/>
            ":/wwww3.orgTR/xhtml1DTD/xhtml1-strict.td>

<html xmlns"http://www.w3..org/1999xhtml xml:lang=en">
<head
<"http:/www.w3.java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 69
<meta java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< =stylesheet/ss="manualcss /
<script src="manual.js" type="text/javascript"></script>
<cript/"overwriteStyle)<cript
</head>
<body class="chap58"  onload==p<id"" nameX80A8E676814A19FD>p>

<div=chlinktop>< ="chap58tmlX80A8E676814A19FD"58span"eadingFields and DivisionRings/span<a

div=""nbsp"/; charset=UTF-/

<span ="ContSS"< />ink"stylesheet"typecss/>
r >span="ocss"nbsp> hrefhtml".12 IsField/</>
<divclassContSS<r/>span=">nbspnbsp/an<a".htmlX871AA7D58263E9AC.-<>/java.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 129
<divspan< =ContSS >pan=nocss&sp;/>hrefchap58html58-3 <//

<<panclass">br /<span ="ocss =ContSS<>class""&&bspspan =hap58htmlX7AA715317A81261B"581- GeneratorsOfField/aspanjava.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 141
spanContSSspan=&&;pan=.#". IsDivisionRinga<span
<span class="ContSS"><br /><span class ="ContSS>
<span>>classnocss;<><"chap58htmlX871AA7D58263E9AC3/>java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129
spanclassspan
< classContSSBlock>
<<panclass="><br >spanclass""> p> id="80A8E676814A19FD="span>
< class">br/>span class=ocss>;&bsp;<span< hrefchap58htmlX7AA715317A81261B>8. GeneratorsOfField//spanjava.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 141
< class"<br/< =nocss>nbsp&;/span>a href"<vclass="<a =chap58htmlX80A8E676814A19FD"> <spanpan""<br/spanclass=">;&;</span<a  hrefchap58htmlX7FE1FA217A08DCE5>.2-Subfield</<span>
<">/<="nbsp>"htmlX7DD27F927BD57FDE>24PrimeFielda<span
</><div
<<classContSS>br/> =">;nbsp;span<ahref"span=ontSSbr< ="nbspnbsp/span>ahref".htmlX86DB31B57FB4F57058- <<a>span
</span
< =ContSS /<<pan=>& classContSS>brspanclass"">nbsp; /spana href=chap58html#>582- RootOfDefiningPolynomial/a/>java.lang.StringIndexOutOfBoundsException: Index 148 out of bounds for length 148
ContSSbr<span=nocssnbspspana="hap58html#FE1FA217A08DCE5"8.1Subfielda/>
<span class="ContSS"><br /><span class=nocssspan =ContSSbr/<=">;;pana "=.X83490C65819D85FE2=< <spannocss &<> href"htmlX8173DA4982DB1E8A"28 a>
<panclassContSSbr >< ="nocss> nbsp;spana =.tmlX86DB31B57FB4F570>58.3PrimitiveElement/a>/>
<span class"<br/><spanclass="nocss&spanclassontSS >span=""nbsp/span =chap58X83490C65819D85FE"58210 <>spanjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
< classContSS / "</>/>
<span=><< ""& <=.tmlX8738C6687D784BB5>.3-<a<span
< class
<spanclass="span=ContSS< > =nocss" />ahref.html">58.-4Norm/><pan
>< > =nocssnbsp;;span>href.htmlX82718B3B818DC699"582-FieldExtension<a<spanjava.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
< class"ntSS<br>span =nocss&bspnbsp;<pan> =hap58html#83490C65819D85FE>..210 //>/>
</div></div>
<div lass"ContSect>< ="ocline< class" </>a =chap58htmlX7D9A02B07D08FA40"58.3< class"">Galois</span/ajava.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 168
</pan
<div class="ContSSBlock">
<span class="ContSS"><span="ContSS"><br >span =/div/>/>
<span=ContSS>br /<<an
<span class classspanclass"ontSS">< /<spanclass=">&bsp
<p; span>< href=chap58X838515278587FF01"583-4Norm/a<spanpan class"ontSSbr/span="">;nbsp/>a ="..tmlX86DB31B57FB4F570>span="""<>spanclassnocss&sp&bspspan< href".htmlX8236A8B47E6AAD9358- NormalBase/spanjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
<a href.htmlX7DD17EB581200AD65835 span="""Tracesof field elements andmatricesspan<<a>
</span>
<spanp class
<span="ContSS">< >span=ocss nbspspan=chap58#8236"583-NormalBase</a>span
</div></div>
<div

<h3> a  <varclass""F<var

<p

<<h458. span=

<p

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

<<>581<span ="Heading"Generating<spanh4>p><em ringeminstrong=pkgstronganontrivial  varArgloisGroup><>

<p><a id

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

<p><a =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

<div class="func">
<A<mdivision ringem <strong""GAP algebraclass>multiplicative and"IsCommutative<code ahref=chap35.#X830A4A4C795FBC2D">spanRefLink.<spana)/>

<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
<pan"GAPprompt>gt;pan< class
<<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
false
true="func">tableclass"func" width"100">trspan="GAPpromptgap;span spanclass""IsDivisionRing ;##  notconstructedasadivision ring</pan>
<span=>gt/ < class"GAPinputmat=[[  ]]  a:=AlgebraRationals[ mat];/pan>
<span
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<>/)
<pcode""><>smallestspan=><span; </>>[< class"Arg><var> ]var =rglistvar>)/<tdclass">&;function;)<td></tableiv

<p><a id="X7D9F7FD4786691EE" name="X7D9F7FD4786691EE"></a></p>

<h5

< class=func>table=func width"pcode class="><codereturns field ="span contains  elementsspan class""><ar =">/>.spanorfield =SimpleMath<spanthat allelementslist < class">/><java.lang.StringIndexOutOfBoundsException: Index 302 out of bounds for length 302
< classfunc> classdiv="">tableclasswidth100">td class=""><code =unc">;DefaultField/> arclass>ist>)< class"&;functionnbsp)td>/>table<div
<p< class="func>DefaultField</ode> returnsafield< class=""></pan  containsalltheelements< =SimpleMath"><ar="">z</ar> .<spanor <span="SimpleMath"><spanspan"GAPprompt"gap<panspan =GAPinputField()Field([( Z()    fieldsspanjava.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126

<

<div
<span="
GF(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
GF(5,GF(6 ="< =""="100%<> classtdleftcodeclass"#8227; Field/>(var=rg"><>]var"listvar)<d>d =tdright>&&bsp/td<tr/<div
<pan"GAPprompt"&;/spanspan"GAPinput>E(9) FieldCF),[ E)   abelian fields/span
CF(9)
AsFieldGaussianRationalsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<panspanclass"GAPprompt>gapt/span><spanclass="">f1 (EB(   ::=efaultField 5));</>
NFNF5 NF
CF
<classGAPprompt>>/>< ="GAPinput>  ;  IsSubset(f2 f1)</an
falsejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</><div

<<>aidX7C298A40852C2AFF nameX7C298A40852C2AFF>a>/p

<h5>58.1-5 DefaultFieldByGenerators</h5

div=func class"width"00"tr<tdclass=tdleft> class=">#;DefaultFieldByGenerators([varArg/varArg./>/< =tdright;operation<>/
<p>returns the defaultpreturns defaultfieldcontainingthe <var="Arg"><var> =GAPprompt>ap;/pan> < =<eturns default containingelementsvarclassArg> < classSimpleMath<>This maybebiggerthesmallestcontainingelementspjava.lang.StringIndexOutOfBoundsException: Index 195 out of bounds for length 195

<p><a id="X7EF624958648D0FA" name="X7EF624958648D0FAjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<<<>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<>)</

<p><a id="X8641861A8550F8BE" name="X8641861A8550F8BE"></a><iv class""><table

<h5

<div classdiv classfunc
<divclassfunctable classfunc"100">
<p>alledwith a field< class="Arg">F</> and

<p><code class

<>aid

h5AsDivisionRing>

< the="> ""="%>tr =tdleft =""8227GeneratorsOfField>classF> )td classnbsp)>tr<iv
<div=<
<pIn secondjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<> with var=Arg<>>andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><code class="func">AsField</code> is just

<p><a id

<4>

<p>aid

<h5h5>21  Subfield>

classfunc>fthe <arclass""<varbea division code=ode(<varvarcodeis division consists elementsvar"/>, as avector    fieldotherwise<ode class"eywfail> returned/>
<iv="table class""width"100>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<<a

<p><a id="X82A0E79A7B9799E0<

<<>< classunc>tableclass" widthp>hisoperationisneededfor   of codeclass=">Field/>(href"hap58htmlX871AA7D58263E9AC"span=RefLink.-<pan< or codeclassFieldByGenerators> ahref#X8641861A8550F8BE class"RefLink">8<>a)  subfieldorder construct  leftacting fora fieldp>

<div
<> operation needed  of< class">

<> =X86DB31B57FB4F570 nameX86DB31B57FB4F570/a<>

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

<div class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>an of <varclassrgD/varthat<arclassrgD<> as division a  ring with leftdomain

<pConstructs the<>Theem>primefield<em><p>  elementof<class"Arg"<>that generatesar="rg"/>asdivisionwith  actingdomainp>

<p>aidjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class="func" width="100%"><
<pThe> <

<<The<> <emofdivision < =">D/>>the  field   contained <class""D</var>.or ,theof  in  isisomorphic the of  .<p

<h5>58.2-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<iv="func"
<pA  class"func> class="func

<><a id"X7845CECE86A83219"="X7845CECE86A83219>/>/>

<h5>58.2-6 DegreeOverPrimeField

<=""
<p<

<The>s ofvar""/over primesee< class><code( hrefhap58html#"<pan class"efLink./span<a)./p>

<<<>58-<

<=
<pis defining ofthefield <var="Arg"></> a field over the actingdomainolynomial/code(<var=rgvar )</tdd class"tdright">(nbsp;ttribute;<td<tr<table/>

<p><a idp> ="X82718B3B818DC699p> idX7845CECE86A83219 name="7845CECE86A83219>>p

<h5.-8 RootOfDefiningPolynomial</h5>

< classtable"func" width%<< ""> =func8227 </code(var"">/ar> class>&;&;<d/>/>/iv
<p>is a root in the field <var class="Arg">F</var> ofits  polynomial a extensionover left  domain var=ArgFvar> the obtained  aroot irreducible var"olyar    < classArg>F/ar><

<<

<h5>58.2-9

<div class>is>58.  class>s  field adjoining arootof  irreduciblepolynomialvarvarpjava.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145
< field> id"8173DA4982DB1E8A" ="8173"/<<p>

<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>

<ivclass="func"><table="func" width"00"><trtd=tdleftcodeclassfunc&8227;Subfields<code(< classArg
<p> thejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0

<p><ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>

<

<>he<emnormem is

<Thus

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

<p>.3<p> normtracespanclass=SimpleMath<spanareof classSimpleMath"<isthe  allsubfields  field<arclass"ArgF/.pjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

<

<p>Note that <code class

<>

<p>or elementsinjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p<>If<span class="SimpleMath"L > K</span

<h5p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class="func"
<p> <emGalois <em><otethatfield <spanclassSimpleMath"var="rg/ &;<needm>/ beGaloisp>

< that field <span"var=rg"</>& K<spanneedem<> be Galoisp>

<div
<panclass="GAPprompt>><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class"&;/>< =GAPinput">Size;( g;/span
< >em  var=Fvartheof field var="F/thatfix subfield< class"mpleMath<>classLeftActingDomain class""Fvar )/> pointwise.p>
trueclassAPprompt>&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
< =GAPprompt&;<<span =">( Arg>var gt span need emnot<m>beaGaloisextensionp>
16
true
<<iv

<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<>>
<p>returns the

<divdiv
<span classspanclassAPpromptgapjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 7
=GAPpromptgap;spanspan=GAPinput>(( CFjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
x_1^2+(-E(4))
<spanclass""gap;/spanspan=GAPinputMinimalPolynomial(CF8,E</span>
h5<>/>
</pre>
<p>< =java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>aidX80FE7E017C2D255CnameX80FE7E017C2D255C/p

<p  polynomialis product<=impleMath> )> where="></span  over theconjugatesof<<rclass="Argz/>infield=Arginfieldvar"<var   < ""<>  polynomial returned univariate overvarclass"><> the number ="></>defaulting to )<p>

< ="" class"widthjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<preturns polynomialthattheproduct <classimpleMath(X -)/span  spanclassSimpleMath>/pan runsover theconjugates  <class"rg">z<var thefield <var class=ArgL<var var="</> Thepolynomialisreturnedasa univariatepolynomial  <var classArg"K<> in the numbervar=rginumvar ( to/java.lang.StringIndexOutOfBoundsException: Index 423 out of bounds for length 423

<p>This polynomial is sometimes x_1+-E()< classGAPpromptgap<span< =GAPinput( CF) RationalsE( )<x_12+E4)

<div
<span^+*x_141
x_1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<
x_1
/pre<>

<p><a<div <p>This  sometimes called<>characteristic<em var="><>w..the   < class=impleMath><class"L><classArgK<var.   areinstalled<code classfuncCharacteristicPolynomial/> a =hap24tmlFA0A727CDB060Bspanclass"RefLink>24131<span>/a)  <code="uncTracePolynomialcodein  of </p>

<

< classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><code  class="<pan ="GAPprompt"apgt;<

<p><a id< class <

<h5

< =func<
<ass class""< class"unc ="100">r< class"left class>8227
classfuncTrace> returns  offield<var class"<<var.twofields<ar class"Arg<>andvarclass""K< arethen traceis .  extension <class"impleMath><class"ArgL>&;< class>/var/>,if   field<var=Arg>/> is then<ode""LeftActingDomain <ar="></ar>)/code> is takenasdefaultfor   subfieldvarclass""K</>   fieldis   code ="( =>)codetaken var=">.>>

<p

<p><a

<h5

< class>tablepcode="func>Norm<//> returns thenorm java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<

<p>The listclass=uncwidth%<r<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<divp> resultlist class"func"race>thetrace  thefield < class"/var.If   var class="Arg>Lvarand class"Arg>K<r> given  traceis computed wr.tthe field extensionspanclass"impleMath<lassLvar &; classArgK<var<span only  <var=ArgL<varisthen ="(< class"">L/>)/code is takenas defaultforthesubfieldvar =Arg"K/var,and is  <odeclass"( varclassclass=Arg"emrace4
<class"gt<span> <spanclass=GAPinputrm(8)) );Norm CF,E8));//an>>
1
1
<span class=funcConjugates/code> thelist <em
-E(4)
<span class
-E(java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<<pan=GAPinput(8) ;  Trace(CF8)CF8) E);<span
0
())
<span1
[ classspan="GAPprompt>>span span class="GAPinput"Norm CF(8, CF(4),E8));<span
< classGAPpromptgap&</> <span ="GAPinput">Conjugates(CF(8,CF(4,E(8 ;/>
[ E[ classGAPprompt>&&java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 12
<span class="GAPprompt">gap></span> <span class="GAPinput">Conjugates( CFE(4)4)
[ E(,E(8^E(8 E()^3, 8E()^, -)-(8) java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 62
</pre></div>

<<a ="X8236A8B47E6AAD93" =8236A8B47E6AAD93/</

<h5>/>

<divp>et <var="Arg"F/> bea field is[()-() ]
<>

<pIfsecond <

<div <   <var=">/var> bea    Galois extensionof subfieldcode =" < ="Argvar <code>Then codeclass=ncNormalBasecodereturns     <varclass="">F</var thatforma normalbasisof <class"rgar that,avectorspacebasisthatisclosed  ofthegroupsee<odeclass""aloisGroupcode ( =chap58#CAA5BA82F09ED2span=RefLink58.<span<>)  <var class"F/var>/>
< ="GAPprompt">gap></> < class<spanclass"GAPprompt"
[ -E(5), -(5)^, -E(5)3, -E(5)^4 ]
<span <span class="h5>58.--NormalBaseh5>
[ 1/2-1/2*< classfunc< class =[/212E4 1/12E4) ]
an
[ Z([ -(
<>  second varArg</var>is, itis ashintfor  to<pan[3)2 (^)^ Z3^)^,Z36),,36),,Z3^ ]
[ (2^6) Z 1/-*4)/+1/2*E()
vjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of<span,-

<ivclasschlinkbot

<hr /ivclass"chlinkbot">spanjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
<p class="foot" />
</body>
</tml

Messung V0.5 in Prozent

¤ Dauer der Verarbeitung: 0.8 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.