{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 0 "" }{TEXT 258 88 "Maple aan het werk met thuisopgave 2 (update van de Map leV5 versie naar Maple8 formaat) " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "Deze keer is alle relevante Maple-uitvoer vermeld; uw computer is inmiddels snel genoeg. Ervaar wat Maple allemaal kan, met behulp van \+ het package \"GF\" (Galois Field = Eindig Lichaam). Het package met r eadlib(GF) laden is niet meer nodig." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 30 "M.b.v. ConvertIn en ConvertOut" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "L:=GF(5,2,delta^2+delta+2): \+ d:=L[ConvertIn](delta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGd&%&d eltaG\"\"&!\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Hier zien we da t de vroegere expliciete getalswaarde d=10^4 (of hoger wanneer bij voo rbeeld de karakteristiek hoger is) van de interne representatie van " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 141 " niet langer aan d e gebruiker meegedeeld wordt. De ConvertIn operator geeft nu de schrij fwijze als lineaire combinatie van de standaardbasis\{" }{XPPEDIT 18 0 "1,delta;" "6$\"\"\"%&deltaG" }{TEXT -1 11 "\}. Nu naar " }{TEXT 262 10 "oplopende " }{TEXT -1 12 "machten van " }{XPPEDIT 18 0 "delta; " "6#%&deltaG" }{TEXT -1 10 " geordend." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "L[ConvertIn](delta^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#d&%&deltaG\"\"&$%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " for i from 0 to 24 do\n delta^i = L[ConvertOut](L[`^`](d,i))\nod;" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&deltaGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*$)%&deltaG\"\"#\"\"\",&*&\"\"%F(F&F(F(\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"$\"\"\",&*&\"\"%F(F&F(F(\"\"#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"%\"\"\",&*&\"\"$F(F&F( F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"&\"\"\",& *&\"\"%F(F&F(F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\" \"'\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"(\" \"\",$*&\"\"#F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG \"\")\"\"\",&*&\"\"$F(F&F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* $)%&deltaG\"\"*\"\"\",&*&\"\"$F(F&F(F(\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#5\"\"\",&F&F(\"\"%F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*$)%&deltaG\"#6\"\"\",&*&\"\"$F(F&F(F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#7\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#8\"\"\",$*&\"\"%F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#9\"\"\",&F&F(\"\"#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#:\"\"\",&F&F(\"\"$F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#;\"\"\",&*&\"\"#F(F&F( F(\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#<\"\"\",&F &F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#=\"\"\"\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#>\"\"\",$*&\"\"$F(F &F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#?\"\"\",&*&\" \"#F(F&F(F(\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#@ \"\"\",&*&\"\"#F(F&F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&d eltaG\"#A\"\"\",&*&\"\"%F(F&F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*$)%&deltaG\"#B\"\"\",&*&\"\"#F(F&F(F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#C\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Let op de andere volgorde in de externe representatie." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "L[isPrimitiveElement](d); " }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 303 " Elke externe (\"wiskundige\") bewerking heeft intern zijn tegenhanger. Bij voorbeeld de bewerking machtsverhe ffen, notatie ^ , vinden we intern terug in de operator L[`^`], waarb ij L de gekozen naam van ons lichaam is. Voor een overzicht zie het \" topic\" GF in de helpfunctie. Aan de uitvoer zien we dat " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 50 "^6 = 2 in L (het is natuurlij k geen verassing dat " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 224 "^6 in Z mod 5 ligt, want zijn vierde macht is gelijk aan 1 en i.v .m. de kleine stelling van Fermat kennen we al alle oplossingen in L v an de vergelijking. x^4=1 (hier is thm.15.8.2 of thm.13.9(ii) toepasb aar). Het feit dat " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 64 "^6 =2 reduceert het handmatig bepalen van de tabel aanzienlijk: " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 14 "^(6*i+j)=2^i *" } {XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 466 "^j ; we kunnen het \+ rekenen vrijwel beperken tot het geval j=0..5. Hier onder volgt nu een aangepaste versie van ons vroegere \"samengestelde\" voorbeeld. De aa npassing was nodig omdat getallen als 4 en 1 in L (voorkomend in de ve rmenigvuldiging en de optelling in het voorbeeld) eerst vervangen moet en worden door hun interne representaties. Dit kan zowel met de L[inpu t]-operator of met de L[ConvertIn]-operator. De nu overbodige L[Conver tOut]-operator is weggelaten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "(delta^4)^26*(4*delta+1) = L[`*`](L[`^`](L[`^`](d,4),26),L[`*`] (L[input](4),L[`+`](d,L[ConvertIn](1))));" }{TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&)%&deltaG\"$/\"\"\"\",&*&\"\"%F(F&F(F(F(F(F(d &F&\"\"&!%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 212 "Stap voor stap de \+ bewerkingen laten uitvoeren lijkt dus handiger (anders vergeet je al s nel een haakje). Enkele speciale gevallen volgen hieronder. In strijd \+ met wat in de helpfunctie staat is de oorspronkelijke " }{TEXT 263 10 "afwijkende" }{TEXT -1 86 " betekenis van L[0] en L[1] als nul- en een element van het lichaam geschrapt. L[i] is " }{TEXT 264 6 "altijd" } {TEXT -1 71 " het ide element van de bijv. lijst L. Nogmaals:L[`-`](0, 1) werkt niet!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "L[`-`](L[ input](4)); n:=L[ConvertIn](0); e:=L[input](1);L[`-`](n,e);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#d%%&deltaG\"\"&\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGd$%&deltaG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGd%%&deltaG\"\"&\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#d%%&deltaG\"\"&%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 260 21 "M.b.v.alias en Ro otOf" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 318 "In afwijking met het vori ge moeten we nu steeds de karakterteristiek blijven specificeren m.b.v .\"mod 5\". N.B. mod5 zonder spatie tussen mod en 5 werkt niet. Deze p aragraaf is vrijwel ongewijzigd. Alleen het verouderde en overbodige r eadlib(GF) is weggelaten omdat vanaf versie 6 het \"GF package\" direc t beschikbaar is." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restar t:alias(delta=RootOf(x^2+x+2)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i from 0 to 24 do\n delta^i = Normal(delta^i)mod 5;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&deltaGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&de ltaG\"\"#\"\"\",&*&\"\"%F(F&F(F(\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"$\"\"\",&*&\"\"%F(F&F(F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"%\"\"\",&*&\"\"$F(F&F(F(\"\"#F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"&\"\"\",&*&\"\"%F(F&F (F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"'\"\"\"\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"(\"\"\",$*&\"\"#F (F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\")\"\"\",&*& \"\"$F(F&F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"\"* \"\"\",&*&\"\"$F(F&F(F(\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %&deltaG\"#5\"\"\",&F&F(\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$ )%&deltaG\"#6\"\"\",&*&\"\"$F(F&F(F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#7\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *$)%&deltaG\"#8\"\"\",$*&\"\"%F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#9\"\"\",&F&F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#:\"\"\",&F&F(\"\"$F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*$)%&deltaG\"#;\"\"\",&*&\"\"#F(F&F(F(\"\"$F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#<\"\"\",&F&F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#=\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#>\"\"\",$*&\"\"$F(F&F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#?\"\"\",&*&\"\"#F(F&F(F (\"\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#@\"\"\",&*& \"\"#F(F&F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&deltaG\"#A \"\"\",&*&\"\"%F(F&F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%&d eltaG\"#B\"\"\",&*&\"\"#F(F&F(F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*$)%&deltaG\"#C\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Factor(x^5-x)mod 5;\nQ:=simplify((x^25-x)/(x^5-x));\nFactor(Q)mod \+ 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&%\"xG\"\"\"\"\"$F&F&,&F%F&\" \"%F&F&,&F%F&\"\"#F&F&,&F%F&F&F&F&F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG,.*$)%\"xG\"#?\"\"\"F**$)F(\"#;F*F**$)F(\"#7F*F**$)F(\"\") F*F**$)F(\"\"%F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*6,(*$)%\"xG \"\"#\"\"\"F)*&\"\"$F)F'F)F)\"\"%F)F),(F%F)*&F(F)F'F)F)F+F)F),(F%F)F'F )F(F)F),(F%F)*&F,F)F'F)F)F(F)F),&F%F)F+F)F),(F%F)*&F+F)F'F)F)F+F)F),(F %F)*&F,F)F'F)F)F)F)F),(F%F)F'F)F)F)F),(F%F)*&F(F)F'F)F)F,F)F),&F%F)F(F )F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "irr:=[x^2+x+2,x^2+3, x^2+4*x+2,x^2+2*x+3,x^2+x+1,x^2+3*x+3,x^2+2,x^2+3*x+4,x^2+4*x+1,x^2+2* x+4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$irrG7,,(*$)%\"xG\"\"#\"\" \"F+F)F+F*F+,&F'F+\"\"$F+,(F'F+*&\"\"%F+F)F+F+F*F+,(F'F+*&F*F+F)F+F+F- F+,(F'F+F)F+F+F+,(F'F+*&F-F+F)F+F+F-F+,&F'F+F*F+,(F'F+*&F-F+F)F+F+F0F+ ,(F'F+*&F0F+F)F+F+F+F+,(F'F+*&F*F+F)F+F+F0F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "for i from 1 to 10 do\n irr[i] = Factor(irr[i],R ootOf(x^2+x+2))mod 5;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)% \"xG\"\"#\"\"\"F)F'F)F(F)*&,(F'F)%&deltaGF)F)F)F),&F'F)*&\"\"%F)F,F)F) F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"F)\"\"$F)* &,(F'F)%&deltaGF)F*F)F),(F'F)*&\"\"%F)F-F)F)F(F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"%F)F'F)F)F(F)*&,&F'F)%& deltaGF)F),(F'F)*&F+F)F.F)F)F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,(*$)%\"xG\"\"#\"\"\"F)*&F(F)F'F)F)\"\"$F)*&,&F'F)*&F+F)%&deltaGF)F)F ),(F'F)*&F(F)F/F)F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\" xG\"\"#\"\"\"F)F'F)F)F)*&,(F'F)*&F(F)%&deltaGF)F)\"\"%F)F),(F'F)*&\"\" $F)F-F)F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\" \"\"F)*&\"\"$F)F'F)F)F+F)*&,&F'F)*&F(F)%&deltaGF)F)F),(F'F)*&F+F)F/F)F )F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"#\"\"\"F)F(F )*&,(F'F)*&\"\"$F)%&deltaGF)F)\"\"%F)F),(F'F)*&F(F)F.F)F)F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"$F)F'F)F )\"\"%F)*&,(F'F)%&deltaGF)F(F)F),(F'F)*&F,F)F/F)F)F)F)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"%F)F'F)F)F)F)*&,(F 'F)*&\"\"$F)%&deltaGF)F)F)F)F),(F'F)*&F(F)F0F)F)F/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&F(F)F'F)F)\"\"%F)*&,(F'F )*&F+F)%&deltaGF)F)\"\"$F)F),(F'F)F/F)F+F)F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 81 "Nu demonstreren we even het verschil tussen expand, exp and mod 5 en Expand mod 5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "(x+delta+4)*(x+4*delta+3) = expand((x+delta+4)*(x+4*delta+3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(%\"xG\"\"\"*&\"\"%F'%&deltaGF'F' \"\"$F'F',(F&F'F*F'F)F'F',.*$)F&\"\"#F'F'*(\"\"&F'F*F'F&F'F'*&\"\"(F'F &F'F'*&F)F')F*F0F'F'*&\"#>F'F*F'F'\"#7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "(x+delta+4)*(x+4*delta+3) = expand((x+delta+4)*(x+4*d elta+3))mod 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(%\"xG\"\"\"*&\" \"%F'%&deltaGF'F'\"\"$F'F',(F&F'F*F'F)F'F',,*$)F&\"\"#F'F'*&F0F'F&F'F' *&F)F')F*F0F'F'*&F)F'F*F'F'F0F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "(x+delta+4)*(x+4*delta+3) = Expand((x+delta+4)*(x+4*delta+3))m od 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(%\"xG\"\"\"*&\"\"%F'%&de ltaGF'F'\"\"$F'F',(F&F'F*F'F)F'F',(*$)F&\"\"#F'F'*&F0F'F&F'F'F)F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Idem met simplify:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "(x-delta^22)*(x-delta^14)=simplify (expand(x-delta^22)*(x-delta^14));(x-delta^22)*(x-delta^14)=simplify(e xpand(x-delta^22)*(x-delta^14) mod 5);\n(x-delta^22)*(x-delta^14)=simp lify(Expand(x-delta^22)*(x-delta^14) mod 5);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"#AF'!\"\"F',&F&F'*$)F*\"#9 F'F,F',,*$)F&\"\"#F'F'*&\"&:\\)F'F*F'F'*(\"%]9F'F*F'F&F'F'*&\"$s&F'F&F 'F'\"'9$z#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&d eltaG\"#AF'!\"\"F',&F&F'*$)F*\"#9F'F,F',,*$)F&\"\"#F'F'*&\"(S'e8F'F*F' F'*(\"%+eF'F*F'F&F'F,*&\"%)G#F'F&F'F,\"(C!pWF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"#AF'!\"\"F',&F&F'*$)F*\"#9 F'F,F',,*$)F&\"\"#F'F'*(\"$l$F'F*F'F&F'F'*&\"#7F'F&F'F'*&\"%+6F'F*F'F' \"$'pF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "(x-delta^22)*(x- delta^14)=simplify(Expand(x-delta^22)*(x-delta^14) mod 5)mod 5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"#AF'!\"\" F',&F&F'*$)F*\"#9F'F,F',(*$)F&\"\"#F'F'*&F4F'F&F'F'\"\"%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "(x-delta^22)*(x-delta^14)=simplify( (x-delta^22)*(x-delta^14))mod 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &,&%\"xG\"\"\"*$)%&deltaG\"#AF'!\"\"F',&F&F'*$)F*\"#9F'F,F',(*$)F&\"\" #F'F'*&F4F'F&F'F'\"\"%F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Nu vo lgt ons onvolprezen ringautomorfisme " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 24 ". In de literatuur heet " }{XPPEDIT 18 0 "mu;" "6#%#muG " }{TEXT -1 36 " meestal het Frobenius automorfisme." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mu:= t -> t^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*$)9$\"\"&\"\" \"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "m:= [seq(5*i mo d 24,i=1..23)]; \nwhattype(m); m[22];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG79\"\"&\"#5\"#:\"#?\"\"\"\"\"'\"#6\"#;\"#@\"\"#\"\"(\"#7\"#< \"#A\"\"$\"\")\"#8\"#=\"#B\"\"%\"\"*\"#9\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%listG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "for i from 1 to 23 do\n (x -delta^i)*(x-delta^(m[i])) = simplify((x-delta^i)*(x-delta^(m[i])))m od 5;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"%&deltaG !\"\"F',&F&F'*$)F(\"\"&F'F)F',(*$)F&\"\"#F'F'F&F'F1F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"\"#F'!\"\"F',&F&F'*$) F*\"#5F'F,F',(*$)F&F+F'F'*&\"\"$F'F&F'F'\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"\"$F'!\"\"F',&F&F'*$)F*\"# :F'F,F',&*$)F&\"\"#F'F'F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&% \"xG\"\"\"*$)%&deltaG\"\"%F'!\"\"F',&F&F'*$)F*\"#?F'F,F',(*$)F&\"\"#F' F'*&F+F'F&F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\" %&deltaG!\"\"F',&F&F'*$)F(\"\"&F'F)F',(*$)F&\"\"#F'F'F&F'F1F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%\"xG\"\"\"*$)%&deltaG\"\"'F(!\" \"\"\"#F(,(*$)F'F.F(F(\"\"%F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ *&,&%\"xG\"\"\"*$)%&deltaG\"\"(F'!\"\"F',&F&F'*$)F*\"#6F'F,F',(*$)F&\" \"#F'F'*&F4F'F&F'F'\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\" xG\"\"\"*$)%&deltaG\"\")F'!\"\"F',&F&F'*$)F*\"#;F'F,F',(*$)F&\"\"#F'F' F&F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&delta G\"\"*F'!\"\"F',&F&F'*$)F*\"#@F'F,F',&*$)F&\"\"#F'F'F4F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"#5F'!\"\"F',&F&F'*$ )F*\"\"#F'F,F',(*$)F&F0F'F'*&\"\"$F'F&F'F'\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"\"(F'!\"\"F',&F&F'*$)F*\"# 6F'F,F',(*$)F&\"\"#F'F'*&F4F'F&F'F'\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%\"xG\"\"\"*$)%&deltaG\"#7F(!\"\"\"\"#F(,(*$)F'F. F(F(*&F.F(F'F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\" \"*$)%&deltaG\"#8F'!\"\"F',&F&F'*$)F*\"#F '!\"\"F',&F&F'*$)F*\"#BF'F,F',(*$)F&\"\"#F'F'*&\"\"$F'F&F'F'F6F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"#?F'!\"\" F',&F&F'*$)F*\"\"%F'F,F',(*$)F&\"\"#F'F'*&F0F'F&F'F'F'F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"*$)%&deltaG\"\"*F'!\"\"F',&F&F'* $)F*\"#@F'F,F',&*$)F&\"\"#F'F'F4F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&,&%\"xG\"\"\"*$)%&deltaG\"#9F'!\"\"F',&F&F'*$)F*\"#AF'F,F',(*$)F&\" \"#F'F'*&F4F'F&F'F'\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\" xG\"\"\"*$)%&deltaG\"#>F'!\"\"F',&F&F'*$)F*\"#BF'F,F',(*$)F&\"\"#F'F'* &\"\"$F'F&F'F'F6F'" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 21 " [input] en [output]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:g:= 29^4-1;g/16;ifactor(g);ifactor(44205);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"'!G2(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&0U%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*,)-%!G6#\"\"#\"\"%\"\"\"-F&6#\"\"$F*- F&6#\"\"&F*-F&6#\"\"(F*-F&6#\"$@%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #**-%!G6#\"\"$\"\"\"-F%6#\"\"&F(-F%6#\"\"(F(-F%6#\"$@%F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "We nemen nu een wat groter voorbeeld dan \+ in het oorspronkelijke geval" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "K:=GF(29,4,alpha^4+alpha+7): a:=K[ConvertIn](alpha); K[Convert In](alpha^4);K[ConvertIn](alpha^44205);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGd&%&alphaG\"#H!\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#d&%&a lphaG\"#H7=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#d%%&alphaG\"#H\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "K[output](a);K[output](K[`+` ](a,K[ConvertIn](1)));K[output](K[`^`](a,44205));K[input](29);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#d&%&alphaG\"#H!\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "K[isPrimitiveElement](a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "En dat klopt de orde van " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 0 "" }{TEXT -1 120 " is niet g maar g/16. Met de methode van het werkco llege zullen we aantonen dat de orde van alpha een deler van g/4 is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "for i from 1 to 7 do 7^i \+ mod 29 od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "We zien dat de orde van 7 in het onderliggende priemlichaam Z mod 29 \"toeval lig\" gelijk is aan 7, een " }{TEXT 265 5 "echte" }{TEXT -1 0 "" } {TEXT -1 24 " deler van 29-1=28. Dus " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{TEXT -1 68 " is zeker niet primitief in K. Immers, daar het mi nimumpolynoom van " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 26 " ,dit is X^4 +X +7, m.b.v." }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT -1 19 " door toepassen van" }{TEXT -1 1 " " }{XPPEDIT 18 0 "mu; " "6#%#muG" }{TEXT -1 29 " in K te ontbinden is als (X-" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 4 ")(X-" }{XPPEDIT 18 0 "mu;" "6#%#m uG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " ))(X-" }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 3 "^2(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 "))(X-" }{XPPEDIT 18 0 "mu;" "6#%# muG" }{TEXT -1 3 "^3(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 17 ")) of wel als (X-" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 4 ")(X-" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 7 "^29)(X-" } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 11 "^(29^2))(X-" } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 103 "^(29^3)), geldt in \+ het bijzonder [substitueer X-->0, of wel, vergelijk de constante co \353ffici\353nten] dat " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 50 " tot de macht 1+29+29^2+29^3 gelijk is aan 7. Dus " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 43 " tot de macht (29^4-1)/(29-1) =7 en daarmee " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 66 "^(g /4)=7^7 (mod 29)=1 in K. Om te zien dat de feitelijke orde van " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 57 " maar g/16 is, kunne n we nog even Maple wat machten van " }{XPPEDIT 18 0 "alpha;" "6#%&al phaG" }{TEXT -1 22 " in K laten berekenen:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 73 "for i from 0 to 88410 do if K[output](K[`^`](a,i))= 1 then print(i) fi od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"&0U%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&5%))" }}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }