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T\303\251tel." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.1.72. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "8.1.73. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "H1:=subgrel(\{x=[tau]\},D3 ); isnormal(H1);\n" }{MPLTEXT 1 0 48 "H2:=subgrel(\{x=[epsilon]\},D3); isnormal(H2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(subgrelGI(_syslib G6\"6$<#/I\"xGF%7#I$tauGF%-I*grelgroupGF$6$<$F+I(epsilonGF%<%7%F0F0F07 $F+F+7&F0F+F0F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(subgrelGI(_syslibG6\"6$<#/I\"xGF%7 #I(epsilonGF%-I*grelgroupGF$6$<$I$tauGF%F+<%7%F+F+F+7$F0F07&F+F0F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.1.74. 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P\303\251ld\303\241k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 28 "8.1.84. Homomorfizmus magja." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "8.1. 85. Homomorfizmust\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.1.86. Feladat ." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.1.87. F eladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.1 .88. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.1.89. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "*8.1.90. Projekt" }{TEXT 207 8 "\303\255" }{TEXT 207 12 "v cs oportok." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8. 1.91. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 23 "8.1.92. Direkt szorzat." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 54 "8.1.93. V\303\251ges Abel-csoportok alapt\303\251tele." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "*8.1.94. V\30 3\251gesen gener" }{TEXT 207 8 "\303\241" }{TEXT 207 36 "lt Abel-csopo rtok alapt\303\251tele." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.1.95. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 16 "8.1.96. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.1.97. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.1.98. Feladat." }{TEXT 207 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.1.99. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "8.1.100. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 24 "8.1.101. Felad at: diszkr" }{TEXT 207 8 "\303\251" }{TEXT 207 17 "t direkt szorzat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 30 "8.1.102. Cayl ey t\303\251tele." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 43 "8.1.103. Permut\303\241ci\303 \263csoportok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "convert([3,4,2,1,7,6,5],'disjcyc'); convert(%,' permlist',7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7&\"\"\"\"\"$\"\"#\" \"%7$\"\"&\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7)\"\"$\"\"%\"\"#\"\" \"\"\"(\"\"'\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "undefi ne('`&*`'); `&*`:=(x,y)->mulperms(y,x);\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 40 "g:=convert([3,4,2,1,7,6,5],'disjcyc');\n" }{MPLTEXT 1 0 40 "h:=convert([2,5,3,4,1,7,6],'disjcyc');\n" }{MPLTEXT 1 0 33 "f: =g&*h; convert(%,'permlist',7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I \"xG6\"I\"yGF%F%6$I)operatorGF%I&arrowGF%F%-_I&groupG6$%*protectedGI(_ syslibGF%I)mulpermsGF%6$F&F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$ 7&\"\"\"\"\"$\"\"#\"\"%7$\"\"&\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7 $7%\"\"\"\"\"#\"\"&7$\"\"'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$ \"\"\"\"\"%7'\"\"#\"\"(\"\"'\"\"&\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7)\"\"%\"\"(\"\"#\"\"\"\"\"$\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "invperm(f); invperm(h)&*invperm(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\"\"\"\"\"%7'\"\"#\"\"$\"\"&\"\"'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\"\"\"\"\"%7'\"\"#\"\"$\"\"&\"\"'\"\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "S5:=permgroup(5,\{[[1,2]],[[ 1,2,3,4,5]]\}); grouporder(S5); isabelian(S5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG6\"6$\"\"&<$7#7$\"\"\"\"\"#7#7'F+F ,\"\"$\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "H1:=permgroup(5,\{[[1,2,3,4,5]]\}); grouporder(H1); i sabelian(H1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG 6\"6$\"\"&<#7#7'\"\"\"\"\"#\"\"$\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "elements(H1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'7\"7#7'\"\"\"\"\"#\"\"$\"\"%\"\"&7#7'F&F*F)F(F'7# 7'F&F)F'F*F(7#7'F&F(F*F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "gg:=RandElement(S5); H2:=permgroup(5,\{gg\}); grouporder(H2);\n" } {MPLTEXT 1 0 50 "groupmember([[1,2,3,4,5]],H2); groupmember(gg,H2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\"\"\"\"\"#7$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG6\"6$\"\"&<#7$7$\"\"\"\" \"#7$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "in ter(H1,H2); issubgroup(H1,S5); issubgroup(H1,H2);\n" }{MPLTEXT 1 0 50 "isnormal(H1,S5); isnormal(H2,S5); isnormal(H2,H1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "-I*permgroupGI(_syslibG6\"6$\"\"&<\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I& falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protected G" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "8.1. 104. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "parity(g); parity(h); parity(f);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 "8. 1.105.K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} }{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "8.1.106. P\303\251lda." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "[[1,3]]&*[[1,2]]&*[[3,4]]; [[1,2]]&*[[2,1]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.107. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.108. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.109. Fel adat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1. 110. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.111. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.112. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.1.113. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.114. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.1.115. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8.1.116. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "8.1.117. Felad at." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "8.1.118. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 34 "*8 .1.119. Defin\303\255ci\303\263." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "M11:=permgro up(11,\{[[1,2,3,4,5,6,7,8,9,10,11]],[[3,7,11,8],[4,10,5,6]]\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG6\"6$\"#6<$7#7-\" \"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5F'7$7&F-F1F'F27&F.F4F/ F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "L:=[1,2];L[1]:=3;L;" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"$\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "grouporder(M11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%?z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "conjugateclasses:=proc(G) local GG,SS,p,i,f;\n" }{MPLTEXT 1 0 28 " SS:=[]; GG:=elements(G);\n" }{MPLTEXT 1 0 18 " for p in GG do\n" } {MPLTEXT 1 0 15 " f:=false;\n" }{MPLTEXT 1 0 26 " for i to nops( SS) do\n" }{MPLTEXT 1 0 61 " if areconjugate(G,SS[i][1],p) and no t p in SS[i] then\n" }{MPLTEXT 1 0 51 " f:=true; SS[i]:=SS[i] u nion \{p\}; break;\n" }{MPLTEXT 1 0 11 " fi;\n" }{MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 43 " if not f then SS:=[op(SS),\{p\}]; fi; \n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 20 " convert(SS,set);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"GG6\"6'I #GGGF%I#SSGF%I\"pGF%I\"iGF%I\"fGF%F%F%C&>F(7\">F'-_I&groupG6$%*protect edGI(_syslibGF%I)elementsGF%F#?&F)F'I%trueGF4C%>F+I&falseGF4?(F*\"\"\" F=-I%nopsGF46#F(F8@$3-_F2I-areconjugateGF%6%F$&&F(6#F*6#F=F)4-I#inGF46 $F)FHC%>F+F8>FH-I&unionGF46$FH<#F)[@$4F+>F(7$-I#opGF4F@FU-I(convertGF4 6$F(I$setGF4F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "conju gateclasses(M11):nops(%);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "8.1.120. P\303\251lda ." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "S4:=per mgroup(4,\{[[1,2]],[[2,3]],[[3,4]]\}); grouporder(S4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG6\"6$\"\"%<%7#7$\"\"\"\"\"#7# 7$\"\"$F'7#7$F,F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "elements(S4); A4:=permgroup(4,selec t(x->parity(x)=1,%)); grouporder(A4);\n" }{MPLTEXT 1 0 31 "isnormal(A4 ,S4); cosets(S4,A4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<:7\"7#7$\"\"\" \"\"#7$F%7$\"\"$\"\"%7#7$F&F*7#F)7#7%F&F'F*7#7&F&F'F*F+7#7$F'F*7#7&F&F *F+F'7#7&F&F*F'F+7#7&F&F+F*F'7#7&F&F'F+F*7#7&F&F+F'F*7#7%F'F*F+7#7%F&F *F+7#7%F&F+F'7#7%F&F*F'7#7$F'F+7#7%F&F'F+7#7%F'F+F*7$F-FH7#7%F&F+F*7#7 $F&F+7$FQF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG6 \"6$\"\"%<.7\"7$7$\"\"\"\"\"#7$\"\"$F'7#7%F,F-F/7#7%F-F/F'7#7%F,F/F'7# 7%F,F'F-7#7%F,F/F-7#7%F,F-F'7#7%F-F'F/7$7$F,F/7$F-F'7#7%F,F'F/7$7$F,F' 7$F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$7 \"7#7$\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "N1:=per mgroup(4,\{\}); grouporder(N1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*pe rmgroupGI(_syslibG6\"6$\"\"%<\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "N2:=permgroup(4,\{[[1 ,2],[3,4]],[[1,3],[2,4]],[[1,4],[2,3]]\});\n" }{MPLTEXT 1 0 47 "groupo rder(N2); isnormal(N2,A4); cosets(A4,N2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I*permgroupGI(_syslibG6\"6$\"\"%<%7$7$\"\"\"\"\"#7$\"\"$ F'7$7$F+F.7$F,F'7$7$F+F'7$F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%7\"7#7%\"\"#\"\"$\"\"%7#7%F&F(F'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "8.1.121. 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Tov" }{TEXT 207 8 "\303\241" } {TEXT 207 14 "bbi feladatok." }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 42 "8. 2. Gy\305\261r\305\261k \303\251s testek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 " " 0 "" {TEXT 201 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 26 "8.2.1. \+ Megjegyz\303\251s." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" } }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 26 "*8.2.2. Megjegyz\303\251s." } }{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "8.2.3. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "undefine('` &*`'); `&*`:=(x,y)->[x[1]*y[1],x[2]*y[2]];\n" }{MPLTEXT 1 0 13 "[1,0]& *[0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$I)oper atorGF%I&arrowGF%F%7$*&&F$6#\"\"\"F.&F&F-F.*&&F$6#\"\"#F.&F&F2F.F%F%F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!F#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "*8. 2.4. P\303\251lda." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->8.2.5. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->8.2.6. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "->8.2.7. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "*8.2.8. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "8.2.9. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.10. 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" }{TEXT 207 12 "T\303\251tel" }{TEXT 207 1 "." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "X:=\{0,1,2, 3,4\}; map(x->x+x mod 5,X); map(x->x+x+x mod 5,X);\n" }{MPLTEXT 1 0 51 "map(x->x+x+x+x mod 5,X); map(x->x+x+x+x+x mod 5,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"!\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"!\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"!\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'\"\"!\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 9 "8.2.20. G" }{TEXT 207 33 "y \305\261r\305\261 karakterisztik" }{TEXT 207 8 "\303\241" }{TEXT 207 2 "ja" }{TEXT 207 1 "." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%? G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.21. Feladat." }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 53 "8.2.22. R\303\251szgy\305\261r\3 05\261, ide\303\241l." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "8.2.23. P\303\251ld\303\24 1k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "8.2.24. P\303\251ld\303\241k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "8.2.25. P\303\251ld\303\241k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8. 2.26. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.27. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.28. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 18 "->8.2.29. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.30. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.31. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.32. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 9 "*8.2.33. " }{TEXT 207 9 "Reprezent" }{TEXT 207 8 "\303\241" }{TEXT 207 2 "ci" }{TEXT 207 8 "\303\263" }{TEXT 207 2 "k." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.34. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 36 "*8.2.35. Boole-gy\305\261r\305\261k." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.36. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.37. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.38. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.39. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.40. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.41. Felad at." }{TEXT 207 0 "" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.42. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.43. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.44. Felad at." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.45. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 25 "*8 .2.46. Feladat: Stone t" }{TEXT 207 13 "\303\251tele." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 38 "8.2.47. Mell\303\251koszt\30 3\241lyok." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.2.48. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.2.49. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "8.2.50. Faktorgy\305\261r\305\261." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.2. 51. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 27 "*8.2.52. Megjegyz\303\251s." } }{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 28 "8.2.53. Homomorfizmus magja." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.2.54. Homomorfizmus-t\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.2. 55. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.56. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.57. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.58. Fel adat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2. 59. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.60. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.61. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.62. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 23 "8.2.63. Direkt szorzat." }}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8 .2.64. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.2.65. T\303\251tel." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.2.66. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.2.67. Gauss-gy\305\261r\305\261k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.2.68. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 40 "8.2.69. Euklideszi gy \305\261r\305\261k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 37 "8.2.70. \303\201ll\303\255t \303\241s." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 43 "8.2.71. P\303\251lda: Gauss-eg\3 03\251szek." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.72. Feladat." }{TEXT 207 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.73. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.74. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 53 "8.2.75. B\305 \221v\303\255tett euklideszi algoritmus." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "polynomexgcd:=proc(a, b,z) local x0,x1,x2,y0,y1,y2,r0,r1,r2,q;\n" }{MPLTEXT 1 0 43 "x0:=1; y 0:=0; r0:=a; x1:=0; y1:=1; r1:=b;\n" }{MPLTEXT 1 0 4 "do\n" }{MPLTEXT 1 0 38 " if r1=0 then return [x0,y0,r0] fi;\n" }{MPLTEXT 1 0 41 " q: =quo(r0,r1,z); r2:=expand(r0-q*r1);\n" }{MPLTEXT 1 0 45 " x2:=expand( x0-q*x1); y2:=expand(y0-q*y1);\n" }{MPLTEXT 1 0 51 " r0:=r1; x0:=x1; \+ y0:=y1; r1:=r2; x1:=x2; y1:=y2;\n" }{MPLTEXT 1 0 10 "od; end;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"aG6\"I\"bGF%I\"zGF%6,I#x0GF%I#x1G F%I#x2GF%I#y0GF%I#y1GF%I#y2GF%I#r0GF%I#r1GF%I#r2GF%I\"qGF%F%F%C)>F)\" \"\">F,\"\"!>F/F$>F*F7>F-F5>F0F&?(F%F5F5F%I%trueG%*protectedGC-@$/F0F7 O7%F)F,F/>F2-I$quoGF%6%F/F0F'>F1-I'expandGF>6#,&F/F5*&F2F5F0F5!\"\">F+ -FJ6#,&F)F5*&F2F5F*F5FN>F.-FJ6#,&F,F5*&F2F5F-F5FN>F/F0>F)F*>F,F->F0F1> F*F+>F-F.F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "debug(po lynomexgcd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I-polynomexgcdG6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "polynomexgcd(z^3+z+1,z^2+2,z );" }}{PARA 9 "" 1 "" {TEXT 208 50 "\{--> enter polynomexgcd, args = z ^3+z+1, z^2+2, z" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"zG6 \"\"\"$\"\"\"F(F%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,&*$)I\"zG6\"\"\"#\"\"\"F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"F#I\"zG6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"z G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"zG6\"\"\"#\"\"\"F(F 'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"F#I\"zG6 \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"zG6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"zG6 \"!\"\"\"\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"F#I\"zG6\"F#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(\"\"\"F#*$)I\"zG6\"\"\"#F#!\"\"F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"F#I\"zG6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"zG6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"F#I\"zG6\" F#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(\"\"\"F#*$)I\"zG6\"\"\"#F#!\"\"F &F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"\"\"\"$F$*&F#F$I\"zG6\"F$! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"#\"\"$\"\"\"*&#F&F%F&)I\"zG6\"F$F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 ",(*&#\"\"\"\"\"$F%I\"zG6\"F%!\"\"F$F)*&F$F%)F'F&F% F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"F#I\"zG6\"F#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",( \"\"\"F#*$)I\"zG6\"\"\"#F#!\"\"F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"\"#\"\"$\"\"\"*&#F&F%F&)I \"zG6\"F$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&#\"\"\"\"\"$F%I\"zG 6\"F%!\"\"F$F)*&F$F%)F'F&F%F)" }}{PARA 9 "" 1 "" {TEXT 208 62 "<-- exi t polynomexgcd (now at top level) = [1+z, 1-z^2-z, 3]\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%,&\"\"\"F$I\"zG6\"F$,(F$F$*$)F%\"\"#F$!\"\"F%F+\" \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "undebug(polynomexgcd );" }}{PARA 11 "" 1 "" {XPPMATH 20 "I-polynomexgcdG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "polynomexgcd(z^3+z+1,z^2+2,z);" }} {PARA 11 "" 1 "" {XPPMATH 20 "7%,&\"\"\"F$I\"zG6\"F$,(F$F$*$)F%\"\"#F$ !\"\"F%F+\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.2.76. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.2.77. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.2 .78. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" } }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.2.79. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 39 "*8.2.80. Maxim\303\2 41lis ide\303\241l." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 36 "*8.2.81. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.2.82. T\303\251t el." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 33 "*8.2.83. Pr\303\255mide\303\241l." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.2.84. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 36 "*8.2 .85. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.2.86. T\303 \251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "8.2.87. H\303\241nyadostest." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "r1 :=(z^3-1)/(z^2-1); r1:=simplify(r1); r2:=simplify((z^4-1)/(z^3-1));\n" }{MPLTEXT 1 0 6 "r1*r2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*$)I\"zG 6\"\"\"$\"\"\"F)F)!\"\"F),&*$)F&\"\"#F)F)F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*$)I\"zG6\"\"\"#\"\"\"F)F&F)F)F)F),&F)F)F&F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&,**$)I\"zG6\"\"\"$\"\"\"F)*$)F&\"\"#F) F)F&F)F)F)F),(F*F)F&F)F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,& \"\"\"F$I\"zG6\"F$!\"\",**$)F%\"\"$F$F$*$)F%\"\"#F$F$F%F$F$F$F$" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.2.88. K\303\266vetkezm\303\251ny." }{TEXT 207 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 43 "*8.2.89. Algebrai strukt\30 3\272r\303\241k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.90. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.91. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.92. Fela dat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.2.9 3. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 " ->8.2.94. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.95. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.96. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.2.97. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "8.2.98. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "bbi feladatok." }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 14 "8.3. Polinomok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "restart;with(PolynomialTools);" }}{PARA 11 "" 1 "" {XPPMATH 20 "73I0CoefficientListG6\"I2CoefficientVectorGF$I-GcdFreeBas isGF$I?GreatestFactorialFactorizationGF$I(HurwitzGF$I1IsSelfReciprocal GF$I2MinimalPolynomialGF$I0PDEToPolynomialGF$I0PolynomialToPDEGF$I0Shi ftEquivalentGF$I7ShiftlessDecompositionGF$I(ShortenGF$I(ShorterGF$I%So rtGF$I&SplitGF$I'SplitsGF$I*TranslateGF$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "8.3.1. Polinomok." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "p1:=2*x^2+x+3; p2:=5*x^3+9; p1+p2; p1*p2; expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"#\"\"\")I\"xG6\"F$F%F%F'F%\"\"$F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"&\"\"\")I\"xG6\"\"\"$F%F%\"\"*F% " }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"#\"\"\")I\"xG6\"F$F%F%F'F%\" #7F%*&\"\"&F%)F'\"\"$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*&\"\"# \"\"\")I\"xG6\"F%F&F&F(F&\"\"$F&F&,&*&\"\"&F&)F(F*F&F&\"\"*F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 ",.*&\"#5\"\"\")I\"xG6\"\"\"&F%F%*&\"#=F% )F'\"\"#F%F%*&F)F%)F'\"\"%F%F%*&\"\"*F%F'F%F%*&\"#:F%)F'\"\"$F%F%\"#FF %" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 44 "*8.3.2. Form\303\241lis hatv\303\241nyso rok." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->8.3.3. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->8.3.4. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->8.3.5. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->8.3.6. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 39 "8.3.7. Polinom f\303\274ggv\303\251nyek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "map(x->x mod 5,[0,1,2,3,4]); map(x- >x^5 mod 5,[0,1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'\"\"!\"\" \"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'\"\"!\"\"\"\"\"# \"\"$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 66 "8.3.8. A marad\303\251kos oszt\3 03\241s t\303\251tele polinomokra." }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "a:=5*x^4+9; b:=x^2+x+1; r:= rem(a,b,x); q:=quo(a,b,x); expand(q*b+r);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"&\"\"\")I\"xG6\"\"\"%F%F%\"\"*F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(F%F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 ",&\"\"*\"\"\"*&\"\"&F$I\"xG6\"F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"&\"\"\")I\"xG6\"\"\"#F%F%*&F$F%F'F%!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"&\"\"\")I\"xG6\"\"\"%F%F%\"\"*F% " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 95 "8.3.9. K\303\266vetkezm\303\251ny: gy\303\266k t\303\251nyez\305\221 lev\303\241laszt\303\241sa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "quo(x^3-1,x -1,x,'r'); r;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\" \"F(F%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.3.10. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.3. 11. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.3.12. K\303\26 6vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.3.13. K\303\266vetkezm\303 \251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "8.3.14. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "polyd ivisorx:=proc(a,b) local bb,db,r,dr,rr,rrr,q;\n" }{MPLTEXT 1 0 29 "if \+ a=0 then return true fi;\n" }{MPLTEXT 1 0 30 "if b=0 then return false fi;\n" }{MPLTEXT 1 0 37 "bb:=lcoeff(b); db:=degree(b); r:=a;\n" } {MPLTEXT 1 0 31 "while degree(r)>=degree(b) do\n" }{MPLTEXT 1 0 55 " \+ rr:=lcoeff(r); dr:=degree(r); rrr:=irem(rr,bb,'q');\n" }{MPLTEXT 1 0 35 " if rrr<>0 then return false fi;\n" }{MPLTEXT 1 0 31 " r:=expand (r-q*b*x^(dr-db));\n" }{MPLTEXT 1 0 22 "od; evalb(r=0); end;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"aG6\"I\"bGF%6)I#bbGF%I#dbGF%I\"rG F%I#drGF%I#rrGF%I$rrrGF%I\"qGF%F%F%C)@$/F$\"\"!OI%trueG%*protectedG@$/ F&F2OI&falseGF5>F(-I'lcoeffGF56#F&>F)-I'degreeGF5F=>F*F$?(F%\"\"\"FCF% 1F?-F@6#F*C'>F,-FF+FE>F--I%iremGF56%F,F(.F.@$0F-F2F8>F*-I'expandGF 56#,&F*FC*(F.FCF&FC)I\"xGF%,&F+FCF)!\"\"FCFen-I&evalbGF56#/F*F2F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "debug(polydivisorx);" }} {PARA 11 "" 1 "" {XPPMATH 20 "I-polydivisorxG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "polydivisorx(4*x^2-1,2*x+1);" }}{PARA 9 "" 1 "" {TEXT 208 47 "\{--> enter polydivisorx, args = 4*x^2-1, 2*x+1" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"%\"\"\")I\"xG6\"\"\"#F %F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"\"!\"\"*&\"\"#F#I\"xG6\"F#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 9 "" 1 "" {TEXT 208 49 "<-- exit polydivisorx (now at top level) = true \}" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "undebug(polydivisorx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I-polydivisorxG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "polydivisorx(4*x^2-1,2*x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 39 "*8.3.15. Megje gyz\303\251s: pszeudooszt" }{TEXT 207 8 "\303\241" }{TEXT 207 2 "s." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 52 "8.3.16. Megjeg yz\303\251s: Horner-elrendez\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Horner:=proc(L::list,c) local i,LL,r; LL:=[];\n" }{MPLTEXT 1 0 50 "if nops(L)=0 then return L L,0 fi; r:=L[nops(L)];\n" }{MPLTEXT 1 0 36 "for i from nops(L)-1 to 1 \+ by -1 do\n" }{MPLTEXT 1 0 32 " LL:=[r,op(LL)]; r:=L[i]+r*c;\n" } {MPLTEXT 1 0 14 "od; LL,r; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$' I\"LG6\"I%listG%*protectedGI\"cGF&6%I\"iGF&I#LLGF&I\"rGF&F&F&C'>F,7\"@ $/-I%nopsGF(6#F%\"\"!O6$F,F6>F-&F%6#F3?(F+,&F3\"\"\"F>!\"\"F?F>I%trueG F(C$>F,7$F--I#opGF(6#F,>F-,&&F%6#F+F>*&F-F>F)F>F>6$F,F-F&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p:=x^3+4*x^2-3*x+7; Coeffici entList(p,x); Horner(%,2);\n" }{MPLTEXT 1 0 27 "quo(p,x-2,x); rem(p,x- 2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\"xG6\"\"\"$\"\"\"F(*&\" \"%F()F%\"\"#F(F(*&F'F(F%F(!\"\"\"\"(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"(!\"$\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"* \"\"'\"\"\"\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\" \"\"F(*&\"\"'F(F%F(F(\"\"*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#D" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "convert(p,horner);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"(\"\"\"*&,&\"\"$!\"\"*&,&\"\"%F$I \"xG6\"F$F$F,F$F$F$F,F$F$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" " %#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "8.3.17. Megjegyz\303 \251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x^2+1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^# \"\"\",$F#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(x ^2+1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#/I\"xG6\"\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(x^2+1,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msolve(x^2+1,5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$<#/I\"xG6\"\"\"#<#/F%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8 .3.18. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.19. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.20. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 18 "->8.3.21. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.22. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.23. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.24. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.25. Fela dat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.2 6. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 10 " *8.3.27. K" }{TEXT 207 13 "\303\266roszt" }{TEXT 207 8 "\303\241" } {TEXT 207 14 "si polinomok. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 30 "*8.3.28. Wilson t\303\251tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "map(i->(i-1 )!+1 mod i,[i$i=2..30]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7?\"\"!F#\" \"$F#\"\"\"F#F%F%F%F#F%F#F%F%F%F#F%F#F%F%F%F#F%F%F%F%F%F#F%" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 43 "8.3.29. Polinom algebrai deriv\303\241ltja." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p:=x^4+3*x^2+2*x+1; diff(p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,**$)I\"xG6\"\"\"%\"\"\"F(*&\"\"$F()F%\"\"#F(F(*&F,F(F%F(F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"%\"\"\")I\"xG6\"\"\"$F%F%*&\"\"' F%F'F%F%\"\"#F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT 207 27 "*8.3.30. Megjegyz\303\251s." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "8.3.31. T\303 \251tel." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 "8.3 .32. K\303\266vetkezm\303\251ny." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 61 "8.3.33. T\303\266bbsz\303\266r\303\266s gy\303\26 6k\303\266k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.3.34. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 26 "8.3.35. Megjegyz\303\251s." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8. 3.36. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.37. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.38. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 18 "->8.3.39. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 77 "8.3.40. Irreducibilis polinomok \303\251 s testb\305\221v\303\255t\303\251sek." }}{PARA 0 "" 0 "" {TEXT 201 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Split(p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "**,*I\"xG6\"\"\"\"-I'RootOfG6$%*protectedGI(_sysl ibGF%6#,**$)I#_ZGF)\"\"%F&F&*&\"\"$F&)F0\"\"#F&F&*&F5F&F0F&F&F&F&F&-F( 6#,.*$)F0F3F&F&*&F'F&F4F&F&*&,&F3F&*$)F'F5F&F&F&F0F&F&F5F&*&F3F&F'F&F& *$)F'F3F&F&F&-F(6#,.*$F4F&F&*&,&F'F&F7F&F&F0F&F&F3F&F?F&*&F7F&F'F&F&*$ )F7F5F&F&F&F&,&F$F&F7!\"\"F&,&F$F&F'FNF&,&F$F&FDFNF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "p:=x^4; Nextpoly(p,x) mod 2; Nextpoly(%,x ) mod 2; Prevpoly(%,x) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$)I\"x G6\"\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\" \"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\"\"\"F( F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "p:=x^4; Nextprime(p,x) mo d 2; Nextprime(%,x) mod 2; Prevprime(%,x) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$)I\"xG6\"\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ", (*$)I\"xG6\"\"\"%\"\"\"F(F%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(* $)I\"xG6\"\"\"%\"\"\"F(*$)F%\"\"$F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"%\"\"\"F(F%F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "8.3.41. P\303\251ld\303\241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Split(x^2+1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"xG6\"\"\"\"-I'RootOfG6$%*protectedGI(_sysl ibGF%6#,&*$)I#_ZGF)\"\"#F&F&F&F&!\"\"F&,&F$F&F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "p:=x^2+x+1; modpol(x^8+4*x^2,p,x,2); modpo l(1/%,p,x,2);\n" }{MPLTEXT 1 0 19 "modpol(%*%%,p,x,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(F%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.42. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.43. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 18 "->8.3.44. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.45. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.46. Feladat." }{TEXT 207 0 " " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.47. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.48. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.49. Felada t." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.50. F eladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.3. 51. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.52. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.53. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "8.3.54. V\303\251ges testek elemsz" }{TEXT 207 8 "\303\2 41" }{TEXT 207 3 "ma." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "8.3.55 . Megjegyz\303\251sek." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.56. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.57. Alkalmaz" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "s: a Rijndael " }{TEXT 207 27 "\303\251s AES blokkrejtjelz" } {TEXT 207 8 "\305\221" }{TEXT 207 2 "k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "lgn:=8; n:=2^lgn-1; R ijndaelPoly:=Nextprime(Z^lgn,Z) mod 2; alpha:=Z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$b#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I\"ZG6\"\"\")\"\"\"F(*$)F%\"\"%F(F(*$)F%\"\" $F(F(F%F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"ZG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "C:=Matrix([[1,0,0,0,1,1,1,1],[1,1, 0,0,0,1,1,1],[1,1,1,0,0,0,1,1],[1,1,1,1,0,0,0,1],[1,1,1,1,1,0,0,0],[0, 1,1,1,1,1,0,0],[0,0,1,1,1,1,1,0],[0,0,0,1,1,1,1,1]]);c:=Vector([1,1,0, 0,0,1,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedG I(_syslibG6\"6#/I$%idGF'\"*[s\"4<" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I 'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\"*S@?q\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);" }} {PARA 11 "" 1 "" {XPPMATH 20 "7brI#&xG6\"I$AddG6$%*protectedGI(_syslib GF$I(AdjointGF$I3BackwardSubstituteGF$I+BandMatrixGF$I&BasisGF$I-Bezou tMatrixGF$I/BidiagonalFormGF$I-BilinearFormGF$I5CharacteristicMatrixGF $I9CharacteristicPolynomialGF$I'ColumnGF$I0ColumnDimensionGF$I0ColumnO perationGF$I,ColumnSpaceGF$I0CompanionMatrixGF$I0ConditionNumberGF$I/C onstantMatrixGF$I/ConstantVectorGF$I%CopyGF$I2CreatePermutationGF$I-Cr ossProductGF$I-DeleteColumnGF$I*DeleteRowGF$I,DeterminantGF$I)Diagonal GF$I/DiagonalMatrixGF$I*DimensionGF$I+DimensionsGF$I+DotProductGF$I6Ei genConditionNumbersGF$I,EigenvaluesGF$I-EigenvectorsGF$I&EqualGF$I2For wardSubstituteGF$I.FrobeniusFormGF$I4GaussianEliminationGF$I2GenerateE quationsGF$I/GenerateMatrixGF$I2GetResultDataTypeGF$I/GetResultShapeGF $I5GivensRotationMatrixGF$I,GramSchmidtGF$I-HankelMatrixGF$I,HermiteFo rmGF$I3HermitianTransposeGF$I/HessenbergFormGF$I.HilbertMatrixGF$I2Hou seholderMatrixGF$I/IdentityMatrixGF$I2IntersectionBasisGF$I+IsDefinite GF$I-IsOrthogonalGF$I*IsSimilarGF$I*IsUnitaryGF$I2JordanBlockMatrixGF$ I+JordanFormGF$I(LA_MainGF$I0LUDecompositionGF$I-LeastSquaresGF$I,Line arSolveGF$I$MapGF$I%Map2GF$I*MatrixAddGF$I2MatrixExponentialGF$I/Matri xFunctionGF$I.MatrixInverseGF$I5MatrixMatrixMultiplyGF$I+MatrixNormGF$ I,MatrixPowerGF$I5MatrixScalarMultiplyGF$I5MatrixVectorMultiplyGF$I2Mi nimalPolynomialGF$I&MinorGF$I(ModularGF$I)MultiplyGF&I,NoUserValueGF$I %NormGF$I*NormalizeGF$I*NullSpaceGF$I3OuterProductMatrixGF$I*Permanent GF$I&PivotGF$I*PopovFormGF$I0QRDecompositionGF$I-RandomMatrixGF$I-Rand omVectorGF$I%RankGF$I6RationalCanonicalFormGF$I6ReducedRowEchelonFormG F$I$RowGF$I-RowDimensionGF$I-RowOperationGF$I)RowSpaceGF$I-ScalarMatri xGF$I/ScalarMultiplyGF$I-ScalarVectorGF$I*SchurFormGF$I/SingularValues GF$I*SmithFormGF$I*SubMatrixGF$I*SubVectorGF$I)SumBasisGF$I0SylvesterM atrixGF$I/ToeplitzMatrixGF$I&TraceGF$I*TransposeGF$I0TridiagonalFormGF $I+UnitVectorGF$I2VandermondeMatrixGF$I*VectorAddGF$I,VectorAngleGF$I5 VectorMatrixMultiplyGF$I+VectorNormGF$I5VectorScalarMultiplyGF$I+ZeroM atrixGF$I+ZeroVectorGF$I$ZipGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Cinv:=MatrixInverse(C) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedGI(_syslibG6\"6#/I$%idGF'\"*!)>Hp\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S:=proc(x) local i,xx; globa l RijndaelPoly,C,c;\n" }{MPLTEXT 1 0 26 " xx:=convert(x,base,2);\n" } {MPLTEXT 1 0 41 " xx:=add(xx[i]*Z^(i-1),i=1..nops(xx));\n" }{MPLTEXT 1 0 79 " if xx<>0 then\n xx:=modpol(1/xx,RijndaelPoly,Z,2)\n else \n xx:=modpol(" }{MPLTEXT 1 0 20 "xx,RijndaelPoly,Z,2)" }{MPLTEXT 1 0 9 "\n fi;\n" }{MPLTEXT 1 0 30 " xx:=CoefficientList(xx,Z);\n" } {MPLTEXT 1 0 42 " while nops(xx)<8 do xx:=[op(xx),0] od;\n" }{MPLTEXT 1 0 27 " xx:=convert(xx,Vector);\n" }{MPLTEXT 1 0 29 " xx:=Multiply (C,xx) mod 2;\n" }{MPLTEXT 1 0 24 " xx:=Add(xx,c) mod 2;\n" }{MPLTEXT 1 0 30 " add(xx[i]*2^(i-1),i=1..8);\n" }{MPLTEXT 1 0 6 "end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 19 "S(0); S(1); S(2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"6$I\"iGF%I#xxGF%F%F%C+>F(-I(convertG% *protectedG6%F$I%baseGF%\"\"#>F(-I$addGF-6$*&&F(6#F'\"\"\")I\"ZGF%,&F' F8F8!\"\"F8/F';F8-I%nopsGF-6#F(@%0F(\"\"!>F(-I'modpolG6$F-I(_syslibGF% 6&*$F(FF(-FG6&F(FLF:F0>F(-_I0PolynomialToolsGFH I0CoefficientListGF%6$F(F:?(F%F8F8F%2F?\"\")>F(7$-I#opGF-FAFD>F(-F,6$F (I'VectorGFH>F(-I$modGF%6$-_I.LinearAlgebraG6$FIF-I)MultiplyGFH6$I\"CG F%F(F0>F(-F]o6$-_FaoI$AddGFH6$F(I\"cGF%F0-F36$*&F6F8)F0F;F8/F';F8FXF%6 %FLFeoF]pF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$C\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$>\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Sinv:=proc(x) local i,xx; gl obal RijndaelPoly,Cinv,c;\n" }{MPLTEXT 1 0 26 " xx:=convert(x,base,2) ;\n" }{MPLTEXT 1 0 42 " while nops(xx)<8 do xx:=[op(xx),0] od;\n" } {MPLTEXT 1 0 27 " xx:=convert(xx,Vector);\n" }{MPLTEXT 1 0 29 " xx:= Add(xx,c,1,-1) mod 2;\n" }{MPLTEXT 1 0 32 " xx:=Multiply(Cinv,xx) mod 2;\n" }{MPLTEXT 1 0 34 " xx:=add(xx[i]*Z^(i-1),i=1..8);\n" }{MPLTEXT 1 0 55 " if xx<>0 then xx:=modpol(1/xx,RijndaelPoly,Z,2) fi;\n" } {MPLTEXT 1 0 30 " xx:=CoefficientList(xx,Z);\n" }{MPLTEXT 1 0 37 " a dd(xx[i]*2^(i-1),i=1..nops(xx));\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 33 "Sinv(99); Sinv(124); Sinv(119);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"6$I\"iGF%I#xxGF%F%F%C+>F(-I( convertG%*protectedG6%F$I%baseGF%\"\"#?(F%\"\"\"F2F%2-I%nopsGF-6#F(\" \")>F(7$-I#opGF-F6\"\"!>F(-F,6$F(I'VectorG6$F-I(_syslibGF%>F(-I$modGF% 6$-_I.LinearAlgebraG6$FBF-I$AddGFA6&F(I\"cGF%F2!\"\"F0>F(-FE6$-_FII)Mu ltiplyGFA6$I%CinvGF%F(F0>F(-I$addGF-6$*&&F(6#F'F2)I\"ZGF%,&F'F2F2FNF2/ F';F2F7@$0F(F<>F(-I'modpolGFA6&*$F(FNI-RijndaelPolyGF%FinF0>F(-_I0Poly nomialToolsGFAI0CoefficientListGF%6$F(Fin-FY6$*&FfnF2)F0FjnF2/F';F2F4F %6%FdoFVFMF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "X:=proc(x,y,b) local u,v,i,x x,yy; # bitwise xor in b bit length\n" }{MPLTEXT 1 0 49 " xx:=convert (x,base,2); yy:=convert(y,base,2);\n" }{MPLTEXT 1 0 42 " while nops(x x)u+v mod 2,xx, yy);\n" }{MPLTEXT 1 0 30 " add(xx[i]*2^(i-1),i=1..b);\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 11 "X(5,3,4);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"xG6\"I\"yGF%I\"bGF%6'I\"uGF%I\"vG F%I\"iGF%I#xxGF%I#yyGF%F%F%C(>F,-I(convertG%*protectedG6%F$I%baseGF%\" \"#>F--F16%F&F4F5?(F%\"\"\"F:F%2-I%nopsGF26#F,F'>F,7$-I#opGF2F>\"\"!?( F%F:F:F%2-F=6#F-F'>F-7$-FBFGFC>F,-I$zipGI(_syslibGF%6%f*6$F)F*F%6$I)op eratorGF%I&arrowGF%F%-I$modGF%6$,&F)F:F*F:F5F%F%F%F,F--I$addGF26$*&&F, 6#F+F:)F5,&F+F:F:!\"\"F:/F+;F:F'F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Word2Bytes:=proc (x) local xx;\n" }{MPLTEXT 1 0 28 " xx:=convert(x,base,256);\n" } {MPLTEXT 1 0 42 " while nops(xx)<4 do xx:=[op(xx),0] od;\n" }{MPLTEXT 1 0 30 " [xx[4],xx[3],xx[2],xx[1]];\n" }{MPLTEXT 1 0 6 "end;\n" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 62 "Bytes2Word:=proc(x) local i; add (x[i]*256^(4-i),i=1..4) end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 39 "Bytes2Word([0,1,2,3]); Word2Bytes(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"6#I#xxGF%F%F%C%>F'-I(convertG%*protectedG6%F$I%baseGF% \"$c#?(F%\"\"\"F1F%2-I%nopsGF,F&\"\"%>F'7$-I#opGF,F&\"\"!7&&F'6#F5&F'6 #\"\"$&F'6#\"\"#&F'6#F1F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\" xG6\"6#I\"iGF%F%F%-I$addG%*protectedG6$*&&F$F&\"\"\")\"$c#,&\"\"%F.F'! \"\"F./F';F.F2F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&^g'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"!\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 93 "K:=[\"00010203\",\"05060708\",\"0A0B0C0D\",\"0 F101112\"];K:=map(x->convert(x,decimal,hex),K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&Q)000102036\"Q)05060708F$Q)0A0B0C0DF$Q)0F101112F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "7&\"&^g'\")'4\"G%)\"*Th\\o\"\"*'=6FD" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "RijndaelKeys:=proc(K,kk) lo cal i,t,x,tt,k,KK,R; global RijndaelPoly;\n" }{MPLTEXT 1 0 22 " k:=no ps(K); KK:=K;\n" }{MPLTEXT 1 0 9 " R:=1;\n" }{MPLTEXT 1 0 27 " for i from k to kk-1 do\n" }{MPLTEXT 1 0 15 " t:=KK[i];\n" }{MPLTEXT 1 0 33 " if k>6 and (i mod k=4) then\n" }{MPLTEXT 1 0 25 " t:=Wo rd2Bytes(t);\n" }{MPLTEXT 1 0 26 " t:=map(x->S(x),t);\n" } {MPLTEXT 1 0 25 " t:=Bytes2Word(t);\n" }{MPLTEXT 1 0 9 " fi;\n " }{MPLTEXT 1 0 23 " if i mod k=0 then\n" }{MPLTEXT 1 0 25 " t :=Word2Bytes(t);\n" }{MPLTEXT 1 0 33 " t:=[t[2],t[3],t[4],t[1]]; \n" }{MPLTEXT 1 0 26 " t:=map(x->S(x),t);\n" }{MPLTEXT 1 0 33 " \+ tt:=CoefficientList(R,Z);\n" }{MPLTEXT 1 0 45 " tt:=add(tt[i] *2^(i-1),i=1..nops(tt));\n" }{MPLTEXT 1 0 25 " tt:=X(t[1],tt,8); \n" }{MPLTEXT 1 0 31 " t:=[tt,t[2],t[3],t[4]];\n" }{MPLTEXT 1 0 40 " R:=modpol(R*Z,RijndaelPoly,Z,2);\n" }{MPLTEXT 1 0 25 " \+ t:=Bytes2Word(t);\n" }{MPLTEXT 1 0 9 " fi;\n" }{MPLTEXT 1 0 27 " \+ t:=X(t,KK[i-k+1],32);\n" }{MPLTEXT 1 0 21 " KK:=[op(KK),t];\n" } {MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 7 " KK;\n" }{MPLTEXT 1 0 6 "en d;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 20 "RijndaelKeys(K,8);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"KG6\"I#kkGF%6)I\"iGF%I\"tGF%I\"xG F%I#ttGF%I\"kGF%I#KKGF%I\"RGF%F%F%C'>F,-I%nopsG%*protectedG6#F$>F-F$>F .\"\"\"?(F(F,F7,&F&F7F7!\"\"I%trueGF3C'>F)&F-6#F(@$32\"\"'F,/-I$modGF% 6$F(F,\"\"%C%>F)-I+Word2BytesGF%6#F)>F)-I$mapGF36$f*6#F*F%6$I)operator GF%I&arrowGF%F%-I\"SGF%FSF%F%F%F)>F)-I+Bytes2WordGF%FM@$/FE\"\"!C+FJ>F )7&&F)6#\"\"#&F)6#\"\"$&F)6#FH&F)6#F7FN>F+-_I0PolynomialToolsG6$F3I(_s yslibGF%I0CoefficientListGF%6$F.I\"ZGF%>F+-I$addGF36$*&&F+F?F7)F^o,&F( F7F7F:F7/F(;F7-F26#F+>F+-I\"XGF%6%FdoF+\"\")>F)7&F+F\\oF_oFbo>F.-I'mod polGFjo6&*&F.F7F^pF7I-RijndaelPolyGF%F^pF^oFY>F)-F]q6%F)&F-6#,(F(F7F,F :F7F7\"#K>F-7$-I#opGF36#F-F)F-F%6#FgqF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7*\"&^g'\")'4\"G%)\"*Th\\o\"\"*'=6FD\"+\\@T9M\"+L^([Y$\"+7(*o(H$\" +QJ=;M" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "SubBytes:=proc(L) local x,y; global S;\n" }{MPLTEXT 1 0 29 " map(x->map(y->S(y),x),L); \n" }{MPLTEXT 1 0 6 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"LG 6\"6$I\"xGF%I\"yGF%F%F%-I$mapG%*protectedG6$f*6#F'F%6$I)operatorGF%I&a rrowGF%F%-F*6$f*6#F(F%F/F%-I\"SGF%F5F%F%F%F'F%F%F%F$F%6#F7F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "SubBytesinv:=proc(L) local x ,y; global Sinv;\n" }{MPLTEXT 1 0 32 " map(x->map(y->Sinv(y),x),L);\n " }{MPLTEXT 1 0 6 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"LG6 \"6$I\"xGF%I\"yGF%F%F%-I$mapG%*protectedG6$f*6#F'F%6$I)operatorGF%I&ar rowGF%F%-F*6$f*6#F(F%F/F%-I%SinvGF%F5F%F%F%F'F%F%F%F$F%6#F7F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ShiftRow:=proc(L) local b,i, d,LL,LLL; b:=nops(L); LLL:=[];\n" }{MPLTEXT 1 0 34 " if b>6 then d:=1 else d:=0 fi;\n" }{MPLTEXT 1 0 17 " for i to b do\n" }{MPLTEXT 1 0 82 " LL:=[L[i][1],L[1+(i mod b)][2],L[1+(i+1+d mod b)][3],L[1+(i+2+ d mod b)][4]];\n" }{MPLTEXT 1 0 24 " LLL:=[op(LLL),LL];\n" } {MPLTEXT 1 0 12 " od; LLL;\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 66 "ShiftRow([[0,1,2,3],[10,11,12,13],[20,21,22,2 3],[30,31,32,33]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"LG6\"6'I \"bGF%I\"iGF%I\"dGF%I#LLGF%I$LLLGF%F%F%C'>F'-I%nopsG%*protectedGF#>F+7 \"@%2\"\"'F'>F)\"\"\">F)\"\"!?(F(F7F7F'I%trueGF0C$>F*7&&&F$6#F(6#F7&&F $6#,&F7F7-I$modGF%6$F(F'F76#\"\"#&&F$6#,&F7F7-FH6$,(F(F7F7F7F)F7F'F76# \"\"$&&F$6#,&F7F7-FH6$,(F(F7FKF7F)F7F'F76#\"\"%>F+7$-I#opGF06#F+F*F+F% F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&7&\"\"!\"#6\"#A\"#L7&\"#5\"#@ \"#K\"\"$7&\"#?\"#J\"\"#\"#87&\"#I\"\"\"\"#7\"#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ShiftRowinv:=proc(L) local b,i,d,LL,LLL; b:=n ops(L); LLL:=[];\n" }{MPLTEXT 1 0 34 " if b>6 then d:=1 else d:=0 fi; \n" }{MPLTEXT 1 0 17 " for i to b do\n" }{MPLTEXT 1 0 84 " LL:=[L[ i][1],L[1+(i-2 mod b)][2],L[1+(i-3-d mod b)][3],L[1+(i-4-d mod b)][4]] ;\n" }{MPLTEXT 1 0 24 " LLL:=[op(LLL),LL];\n" }{MPLTEXT 1 0 12 " o d; LLL;\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 69 "ShiftRowinv([[0,1,2,3],[10,11,12,13],[20,21,22,23],[30,31,32,33] ]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"LG6\"6'I\"bGF%I\"iGF%I\" dGF%I#LLGF%I$LLLGF%F%F%C'>F'-I%nopsG%*protectedGF#>F+7\"@%2\"\"'F'>F) \"\"\">F)\"\"!?(F(F7F7F'I%trueGF0C$>F*7&&&F$6#F(6#F7&&F$6#,&F7F7-I$mod GF%6$,&F(F7\"\"#!\"\"F'F76#FK&&F$6#,&F7F7-FH6$,(F(F7\"\"$FLF)FLF'F76#F U&&F$6#,&F7F7-FH6$,(F(F7\"\"%FLF)FLF'F76#Fhn>F+7$-I#opGF06#F+F*F+F%F%F %" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&7&\"\"!\"#J\"#A\"#87&\"#5\"\"\"\" #K\"#B7&\"#?\"#6\"\"#\"#L7&\"#I\"#@\"#7\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "normalizepolyzZ:=proc(p)\n" }{MPLTEXT 1 0 31 "loc al i; global RijndaelPoly;\n" }{MPLTEXT 1 0 37 "CoefficientList(expand (p) mod 2,z);\n" }{MPLTEXT 1 0 39 "map(x->modpol(x,RijndaelPoly,Z,2),% );\n" }{MPLTEXT 1 0 45 "add(%[i]*z^(i-1),i=1..nops(%)); sort(%); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"pG6\"6#I\"iGF%F%F%C&-_I0Polyno mialToolsG6$%*protectedGI(_syslibGF%I0CoefficientListGF%6$-I$modGF%6$- I'expandGF-F#\"\"#I\"zGF%-I$mapGF-6$f*6#I\"xGF%F%6$I)operatorGF%I&arro wGF%F%-I'modpolGF,6&F=I-RijndaelPolyGF%I\"ZGF%F6F%F%F%I\"%GF%-I$addGF- 6$*&&FFF&\"\"\")F7,&F'FLFL!\"\"FL/F';FL-I%nopsGF-6#FF-I%sortGF-FTF%6#F DF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "RijndaelMixPoly:=(Z+ 1)*z^3+z^2+z+Z;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 33 "MixMul:=proc (L) local p,LL,x,i;\n" }{MPLTEXT 1 0 40 " global RijndaelPoly,Rijndae lMixPoly;\n" }{MPLTEXT 1 0 36 " LL:=map(x->convert(x,base,2),L);\n" } {MPLTEXT 1 0 50 " LL:=map(x->add(x[i]*Z^(i-1),i=1..nops(x)),LL);\n" } {MPLTEXT 1 0 33 " p:=add(LL[i]*z^(i-1),i=1..4);\n" }{MPLTEXT 1 0 25 " p:=p*RijndaelMixPoly;\n" }{MPLTEXT 1 0 26 " p:=normalizepolyzZ(p); \n" }{MPLTEXT 1 0 27 " while degree(p,z)>=4 do\n" }{MPLTEXT 1 0 72 " \+ p:=normalizepolyzZ(p-lcoeff(p,z)*z^(degree(p,z)-4)*(z^4+1) mod 2); \n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 29 " LL:=CoefficientList( p,z);\n" }{MPLTEXT 1 0 42 " while nops(LL)<4 do LL:=[op(LL),0] od;\n" }{MPLTEXT 1 0 27 " map(x->subs(Z=2,x),LL);\n" }{MPLTEXT 1 0 6 "end; \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 68 "MixMul([1,1,1,1]); MixMul([ 219,19,83,69]); MixMul([242,10,34,92]);\n" }{MPLTEXT 1 0 55 "MixMul([1 98,198,198,198]); MixMul([212,212,212,213]);\n" }{MPLTEXT 1 0 24 "MixM ul([45,38,49,76]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&,&I\"ZG6\"\" \"\"F'F'F')I\"zGF&\"\"$F'F'*$)F)\"\"#F'F'F)F'F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"LG6\"6&I\"pGF%I#LLGF%I\"xGF%I\"iGF%F%F%C+>F(-I$map G%*protectedG6$f*6#F)F%6$I)operatorGF%I&arrowGF%F%-I(convertGF/6%F)I%b aseGF%\"\"#F%F%F%F$>F(-F.6$f*F2F%F3F%-I$addGF/6$*&&F)6#F*\"\"\")I\"ZGF %,&F*FEFE!\"\"FE/F*;FE-I%nopsGF/F2F%F%6$F*8'F(>F'-F@6$*&&F(FDFE)I\"zGF %FHFE/F*;FE\"\"%>F'*&F'FEI0RijndaelMixPolyGF%FE>F'-I0normalizepolyzZGF %6#F'?(F%FEFEF%1FY-I'degreeGF/6$F'FV>F'-Fin6#-I$modGF%6$,&F'FE*(-I'lco effGF/F_oFE)FV,&F]oFEFYFIFE,&*$)FVFYFEFEFEFEFEFIF:>F(-_I0PolynomialToo lsG6$F/I(_syslibGF%I0CoefficientListGF%F_o?(F%FEFEF%2-FM6#F(FY>F(7$-I# opGF/Fip\"\"!-F.6$f*F2F%F3F%-I%subsGF/6$/FGF:F)F%F%F%F(F%6$I-RijndaelP olyGF%FfnF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"F#F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$U\"\"#x\"$h\"\"$)=" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$f\"\"$?#\"#))\"$d\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$)>F#F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$8#F#\"$:#\"$9#" } }{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#x\"$E\"\"$*=\"$[#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "RijndaelMixPolyinv:=(Z^3+Z+1)*z^3+( Z^3+Z^2+1)*z^2+(Z^3+1)*z+Z^3+Z^2+Z;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 36 "MixMulinv:=proc(L) local p,LL,x,i;\n" }{MPLTEXT 1 0 43 " glob al RijndaelPoly,RijndaelMixPolyinv;\n" }{MPLTEXT 1 0 36 " LL:=map(x-> convert(x,base,2),L);\n" }{MPLTEXT 1 0 50 " LL:=map(x->add(x[i]*Z^(i- 1),i=1..nops(x)),LL);\n" }{MPLTEXT 1 0 33 " p:=add(LL[i]*z^(i-1),i=1. .4);\n" }{MPLTEXT 1 0 28 " p:=p*RijndaelMixPolyinv;\n" }{MPLTEXT 1 0 26 " p:=normalizepolyzZ(p);\n" }{MPLTEXT 1 0 27 " while degree(p,z)> =4 do\n" }{MPLTEXT 1 0 72 " p:=normalizepolyzZ(p-lcoeff(p,z)*z^(deg ree(p,z)-4)*(z^4+1) mod 2);\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 29 " LL:=CoefficientList(p,z);\n" }{MPLTEXT 1 0 42 " while nops(LL )<4 do LL:=[op(LL),0] od;\n" }{MPLTEXT 1 0 27 " map(x->subs(Z=2,x),LL );\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 52 " MixMulinv([1,1,1,1]); MixMulinv([142,77,161,188]);\n" }{MPLTEXT 1 0 60 "MixMulinv([159,220,88,157]); MixMulinv([198,198,198,198]);\n" } {MPLTEXT 1 0 60 "MixMulinv([213,213,215,214]); MixMulinv([77,126,189,2 48]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&,(*$)I\"ZG6\"\"\"$\"\"\"F *F'F*F*F*F*)I\"zGF(F)F*F**&,(F%F**$)F'\"\"#F*F*F*F*F*)F,F1F*F**&,&F%F* F*F*F*F,F*F*F%F*F/F*F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"LG6\" 6&I\"pGF%I#LLGF%I\"xGF%I\"iGF%F%F%C+>F(-I$mapG%*protectedG6$f*6#F)F%6$ I)operatorGF%I&arrowGF%F%-I(convertGF/6%F)I%baseGF%\"\"#F%F%F%F$>F(-F. 6$f*F2F%F3F%-I$addGF/6$*&&F)6#F*\"\"\")I\"ZGF%,&F*FEFE!\"\"FE/F*;FE-I% nopsGF/F2F%F%6$F*8'F(>F'-F@6$*&&F(FDFE)I\"zGF%FHFE/F*;FE\"\"%>F'*&F'FE I3RijndaelMixPolyinvGF%FE>F'-I0normalizepolyzZGF%6#F'?(F%FEFEF%1FY-I'd egreeGF/6$F'FV>F'-Fin6#-I$modGF%6$,&F'FE*(-I'lcoeffGF/F_oFE)FV,&F]oFEF YFIFE,&*$)FVFYFEFEFEFEFEFIF:>F(-_I0PolynomialToolsG6$F/I(_syslibGF%I0C oefficientListGF%F_o?(F%FEFEF%2-FM6#F(FY>F(7$-I#opGF/Fip\"\"!-F.6$f*F2 F%F3F%-I%subsGF/6$/FGF:F)F%F%F%F(F%6$I-RijndaelPolyGF%FfnF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"F#F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$>#\"#>\"#$)\"#p" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$U#\"#5\"# M\"##*" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$)>F#F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"$7#F#F#\"$8#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7& \"#X\"#Q\"#\\\"#w" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Rijnda el:=proc(M::list(posint),K::list(posint),r::posint)\n" }{MPLTEXT 1 0 61 " local i,j,k,m,KK,MM,MMM; k:=nops(K); m:=nops(M); MMM:=[];\n" } {MPLTEXT 1 0 32 " KK:=RijndaelKeys(K,m*(r+1));\n" }{MPLTEXT 1 0 17 " \+ for j to m do\n" }{MPLTEXT 1 0 38 " MMM:=[op(MMM),X(KK[j],M[j],32) ];\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 24 " for i from 2 to r \+ do\n" }{MPLTEXT 1 0 36 " MM:=map(x->Word2Bytes(x),MMM);\n" } {MPLTEXT 1 0 23 " MM:=SubBytes(MM);\n" }{MPLTEXT 1 0 23 " MM:=Sh iftRow(MM);\n" }{MPLTEXT 1 0 31 " MM:=map(x->MixMul(x),MM);\n" } {MPLTEXT 1 0 35 " MM:=map(x->Bytes2Word(x),MM);\n" }{MPLTEXT 1 0 14 " MMM:=[];\n" }{MPLTEXT 1 0 19 " for j to m do\n" }{MPLTEXT 1 0 49 " MMM:=[op(MMM),X(KK[(i-1)*m+j],MM[j],32)];\n" }{MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 34 " MM:=ma p(x->Word2Bytes(x),MMM);\n" }{MPLTEXT 1 0 21 " MM:=SubBytes(MM);\n" } {MPLTEXT 1 0 21 " MM:=ShiftRow(MM);\n" }{MPLTEXT 1 0 33 " MM:=map(x- >Bytes2Word(x),MM);\n" }{MPLTEXT 1 0 12 " MMM:=[];\n" }{MPLTEXT 1 0 17 " for j to m do\n" }{MPLTEXT 1 0 43 " MMM:=[op(MMM),X(KK[r*m+j] ,MM[j],32)];\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 8 " MMM;\n" } {MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 59 "M:=[\"5 06812A4\",\"5F08C889\",\"B97F5980\",\"038B8359\"];\n" }{MPLTEXT 1 0 38 "M:=map(x->convert(x,decimal,hex),M);\n" }{MPLTEXT 1 0 19 "Rijndael (M,K,10);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"MG6\"-I%listG%*pr otectedG6#I'posintGF)'I\"KGF&F''I\"rGF&F+6)I\"iGF&I\"jGF&I\"kGF&I\"mGF &I#KKGF&I#MMGF&I$MMMGF&F&F&C/>F3-I%nopsGF)6#F->F4-F;6#F%>F77\">F5-I-Ri jndaelKeysGF&6$F-*&F4\"\"\",&F/FGFGFGFG?(F2FGFGF4I%trueGF)>F77$-I#opGF )6#F7-I\"XGF&6%&F56#F2&F%FT\"#K?(F1\"\"#FGF/FJC)>F6-I$mapGF)6$f*6#I\"x GF&F&6$I)operatorGF&I&arrowGF&F&-I+Word2BytesGF&FinF&F&F&F7>F6-I)SubBy tesGF&6#F6>F6-I)ShiftRowGF&Fco>F6-Ffn6$f*FinF&F[oF&-I'MixMulGF&FinF&F& F&F6>F6-Ffn6$f*FinF&F[oF&-I+Bytes2WordGF&FinF&F&F&F6F@?(F2FGFGF4FJ>F77 $FM-FQ6%&F56#,&*&,&F1FGFG!\"\"FGF4FGFGF2FG&F6FTFVFZF`oFdoF]pF@?(F2FGFG F4FJ>F77$FM-FQ6%&F56#,&*&F/FGF4FGFGF2FGF^qFVF7F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&Q)506812A46\"Q)5F08C889F$Q)B97F5980F$Q)038B8359F$" } }{PARA 11 "" 1 "" {XPPMATH 20 "7&\"+'z(**[8\"+X6T%f\"\"+W487J\")xZZf" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"+fy%*RO\"+@y2!>#\"*7q_7\"\"+t#f1D% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Rijndaelinv:=proc(M::li st(posint),K::list(posint),r::posint)\n" }{MPLTEXT 1 0 61 " local i,j ,k,m,KK,MM,MMM; k:=nops(K); m:=nops(M); MMM:=[];\n" }{MPLTEXT 1 0 32 " KK:=RijndaelKeys(K,m*(r+1));\n" }{MPLTEXT 1 0 17 " for j to m do\n" }{MPLTEXT 1 0 42 " MMM:=[op(MMM),X(KK[r*m+j],M[j],32)];\n" } {MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 34 " MM:=map(x->Word2Bytes(x), MMM);\n" }{MPLTEXT 1 0 24 " MM:=ShiftRowinv(MM);\n" }{MPLTEXT 1 0 24 " MM:=SubBytesinv(MM);\n" }{MPLTEXT 1 0 33 " MM:=map(x->Bytes2Word(x ),MM);\n" }{MPLTEXT 1 0 30 " for i from r to 2 by -1 do\n" }{MPLTEXT 1 0 14 " MMM:=[];\n" }{MPLTEXT 1 0 19 " for j to m do\n" } {MPLTEXT 1 0 49 " MMM:=[op(MMM),X(KK[(i-1)*m+j],MM[j],32)];\n" } {MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 36 " MM:=map(x->Word2Bytes (x),MMM);\n" }{MPLTEXT 1 0 34 " MM:=map(x->MixMulinv(x),MM);\n" } {MPLTEXT 1 0 26 " MM:=ShiftRowinv(MM);\n" }{MPLTEXT 1 0 26 " MM: =SubBytesinv(MM);\n" }{MPLTEXT 1 0 35 " MM:=map(x->Bytes2Word(x),MM );\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 12 " MMM:=[];\n" } {MPLTEXT 1 0 17 " for j to m do\n" }{MPLTEXT 1 0 39 " MMM:=[op(MMM ),X(KK[j],MM[j],32)];\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 8 " \+ MMM;\n" }{MPLTEXT 1 0 6 "end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 60 "CM:=[\"D8F53253\",\"8289EF7D\",\"06B506A4\",\"FD5BE9C9\"];\n" } {MPLTEXT 1 0 40 "CM:=map(x->convert(x,decimal,hex),CM);\n" }{MPLTEXT 1 0 23 "Rijndaelinv(CM,K,10);\n" }{MPLTEXT 1 0 59 "M:=[\"506812A4\",\" 5F08C889\",\"B97F5980\",\"038B8359\"];\n" }{MPLTEXT 1 0 38 "M:=map(x-> convert(x,decimal,hex),M);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\" MG6\"-I%listG%*protectedG6#I'posintGF)'I\"KGF&F''I\"rGF&F+6)I\"iGF&I\" jGF&I\"kGF&I\"mGF&I#KKGF&I#MMGF&I$MMMGF&F&F&C/>F3-I%nopsGF)6#F->F4-F;6 #F%>F77\">F5-I-RijndaelKeysGF&6$F-*&F4\"\"\",&F/FGFGFGFG?(F2FGFGF4I%tr ueGF)>F77$-I#opGF)6#F7-I\"XGF&6%&F56#,&*&F/FGF4FGFGF2FG&F%6#F2\"#K>F6- I$mapGF)6$f*6#I\"xGF&F&6$I)operatorGF&I&arrowGF&F&-I+Word2BytesGF&FinF &F&F&F7>F6-I,ShiftRowinvGF&6#F6>F6-I,SubBytesinvGF&Fco>F6-Ffn6$f*FinF& F[oF&-I+Bytes2WordGF&FinF&F&F&F6?(F1F/!\"\"\"\"#FJC)F@?(F2FGFGF4FJ>F77 $FM-FQ6%&F56#,&*&,&F1FGFGF^pFGF4FGFGF2FG&F6FXFYFZ>F6-Ffn6$f*FinF&F[oF& -I*MixMulinvGF&FinF&F&F&F6F`oFdoFgoF@?(F2FGFGF4FJ>F77$FM-FQ6%&F5FXF[qF YF7F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&Q)D8F532536\"Q)8289EF7DF$ Q)06B506A4F$Q)FD5BE9C9F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"+fy%*RO \"+@y2!>#\"*7q_7\"\"+t#f1D%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"+'z(* *[8\"+X6T%f\"\"+W487J\")xZZf" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&Q)5068 12A46\"Q)5F08C889F$Q)B97F5980F$Q)038B8359F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"+'z(**[8\"+X6T%f\"\"+W487J\")xZZf" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "8.3.58. T\303\251tel." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.3.59. T\303\251tel." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.3.60. K\303\266vetkezm\303\251ny." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "8.3.61. K\303 \266vetkezm\303\251ny." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.62. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.63. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 32 "8.3.64. Irreducibilis polinomok." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "factor(x^3-1); factor(6*x^2+12*x+12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"xG6\"\"\"\"F&!\"\"F&,(*$)F$\"\"#F&F&F$F&F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"'\"\"\")I\"xG6\"\"\"#F%F%*&\"#7F %F'F%F%F+F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "*8.3.65. Primit\303\255v polinom ok." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 30 "*8.3.66. Gauss lemm\303\241ja." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "*8.3.67. Lemma." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 28 "8.3.68. Gauss t\303\251tele." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 30 "*8.3.69. Legnagyobb k\303\266z" }{TEXT 207 14 "\303\266s oszt" }{TEXT 207 8 "\303\263" }{TEXT 207 3 " \+ sz" }{TEXT 207 8 "\303\241" }{TEXT 207 3 "mol" }{TEXT 207 8 "\303\241" }{TEXT 207 10 "sa Gauss-g" }{TEXT 207 18 "y\305\261r\305\261" }{TEXT 207 17 " feletti polinomg" }{TEXT 207 18 "y\305\261r\305\261" }{TEXT 207 15 " eset\303\251n." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.70. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 18 "->8.3.71. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.72. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.73. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.74. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.75. Fela dat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.76. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 51 "*8 .3.77. Sch\303\266nemann-Eisenstein-t\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 36 "*8.3.78. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(x^4+7);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\"\"\"F(\"\"(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "(x^5-1)/(x-1); simplify(%); \+ Translate(%,x,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*$)I\"xG6\"\"\" &\"\"\"F)F)!\"\"F),&F&F)F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I \"xG6\"\"\"%\"\"\"F(*$)F%\"\"$F(F(*$)F%\"\"#F(F(F%F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 ",,\"\"&\"\"\"*&\"#5F$I\"xG6\"F$F$*&F&F$)F'\"\"#F$F $*&F#F$)F'\"\"$F$F$*$)F'\"\"%F$F$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 29 "*8.3.79. Megj egyz\303\251sek." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.80. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "8.3.81. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 44 "8.3.82. Lagrange-interpol\303\241ci\303\263." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "with(CurveFitting);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7*I(BSpline G6\"I-BSplineCurveGF$I,InteractiveGF$I-LeastSquaresGF$I8PolynomialInte rpolationGF$I6RationalInterpolationGF$I'SplineGF$I4ThieleInterpolation GF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "PolynomialInterpolat ion([[0,2],[1,4],[3,7],[4,5]],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",** &#\"\"\"\"\"%F%)I\"xG6\"\"\"$F%!\"\"*&#\"\"&\"\"'F%)F(\"\"#F%F%*&#\"#< \"#7F%F(F%F%F1F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Polynom ialInterpolation([[0,2],[1,4],[3,7],[4,5]],x,form=Lagrange);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",***#\"\"\"\"\"'F%,&I\"xG6\"F%F%!\"\"F%,&F(F% \"\"$F*F%,&F(F%\"\"%F*F%F***#\"\"#F,F%F(F%F+F%F-F%F%**#\"\"(F&F%F(F%F' F%F-F%F***#\"\"&\"#7F%F(F%F'F%F+F%F%" }}}{PARA 0 "" 0 "" {TEXT 201 0 " " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "8.3.83. Titokmegoszt" } {TEXT 207 10 "\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p:=nextprime(10^50);\n" }{MPLTEXT 1 0 56 "t:=12345678901234567890123456789012345678901234567890;\n" } {MPLTEXT 1 0 121 "P:=t+94536789321234890676867542309176345623498734562 349*x+79888998766657439865923165346578934524359870127935*x^2 mod p;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"T^,+++++++++++++++++++++++\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"S!*ycM7!*ycM7!*ycM7!*ycM7!*ycM7" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(\"S!*ycM7!*ycM7!*ycM7!*ycM7!*ycM7\"\" \"*&\"S\\BcM()\\BcMw\"4Banon!*[B@$*yOX*F$I\"xG6\"F$F$*&\"SNz7q)fV_M*yl MlJ#f')Rulm()**)))zF$)F'\"\"#F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y1:=subs(x=1,P) mod p;\n" }{MPLTEXT 1 0 24 "y2:=subs( x=2,P) mod p;\n" }{MPLTEXT 1 0 24 "y3:=subs(x=3,P) mod p;\n" }{MPLTEXT 1 0 24 "y4:=subs(x=4,P) mod p;\n" }{MPLTEXT 1 0 24 "y5:=subs(x=5,P) m od p;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"SB!e#R)fn#einZWkT\"HV)*o7*) pYrn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"StN?%=QL-v2o$z-7b2(3TL5EDv4#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"SU[SpijoKz^d$=dMSr)>c[wNq&\\\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"SI='[4aEc!o!)4drUO_%oJp_k\"oro" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"S'Qu0m\"R0pVn$**>ISD#z,XQn%faA)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Interp([1,3,5],[y1,y3,y5],x) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(\"S!*ycM7!*ycM7!*ycM7!*ycM7 !*ycM7\"\"\"*&\"S\\BcM()\\BcMw\"4Banon!*[B@$*yOX*F$I\"xG6\"F$F$*&\"SNz 7q)fV_M*ylMlJ#f')Rulm()**)))zF$)F'\"\"#F$F$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8 .3.84. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 40 "8.3.85. Kronecker-elj\303\241r\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7CI$ChiG6\"I%bellGF$I)binomialGF$I)car tprodGF$I*characterGF$I'chooseGF$I,compositionGF$I)conjpartGF$I+decode partGF$I+encodepartGF$I*fibonacciGF$I*firstpartGF$I)graycodeGF$I)intto vecGF$I)lastpartGF$I,multinomialGF$I)nextpartGF$I)numbcombGF$I)numbcom pGF$I)numbpartGF$I)numbpermGF$I*partitionGF$I(permuteGF$I)powersetGF$I )prevpartGF$I)randcombGF$I)randpartGF$I)randpermGF$I-setpartitionGF$I* stirling1GF$I*stirling2GF$I(subsetsGF$I)vectointGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I)bigomegaGF$I&cfracGF$I)cfracpolGF$I+cyclot omicGF$I)divisorsGF$I)factorEQGF$I*factorsetGF$I'fermatGF$I)imagunitGF $I&indexGF$I/integral_basisGF$I)invcfracGF$I'invphiGF$I*issqrfreeGF$I' jacobiGF$I*kroneckerGF$I'lambdaGF$I)legendreGF$I)mcombineGF$I)mersenne GF$I(migcdexGF$I*minkowskiGF$I(mipolysGF$I%mlogGF$I'mobiusGF$I&mrootGF $I&msqrtGF$I)nearestpGF$I*nthconverGF$I)nthdenomGF$I)nthnumerGF$I'nthp owGF$I&orderG%*protectedGI)pdexpandGF$I$phiGF$I#piGF$I*pprimrootGF$I)p rimrootGF$I(quadresGF$I+rootsunityGF$I*safeprimeGF$I&sigmaGF$I*sq2fact orGF$I(sum2sqrGF$I$tauGF$I%thueGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Kroneckerfactorx:=proc(p) local d,D,pp,i,X,Y,v,V,c,q, y;\n" }{MPLTEXT 1 0 16 "pp:=expand(p);\n" }{MPLTEXT 1 0 29 "if pp=0 th en return [0] fi;\n" }{MPLTEXT 1 0 28 "if pp=1 then return [] fi;\n" } {MPLTEXT 1 0 31 "if pp=-1 then return [-1] fi;\n" }{MPLTEXT 1 0 36 "D: =degree(pp); X:=[]; Y:=[]; i:=0;\n" }{MPLTEXT 1 0 30 "while nops(X)<=f loor(D/2) do\n" }{MPLTEXT 1 0 20 " v:=subs(x=i,pp);\n" }{MPLTEXT 1 0 68 " if v=0 then return [x-i,op(Kroneckerfactorx(quo(pp,x-i,x)))] fi; \n" }{MPLTEXT 1 0 44 " V:=divisors(v); V:=V union map(y->-y,V);\n" } {MPLTEXT 1 0 48 " V:=[op(V)]; V:=sort(V,(u,v)->abs(u)=2*d do\n" }{MPLTEXT 1 0 27 " y :=cartprod(Y[1..d+1]);\n" }{MPLTEXT 1 0 28 " while not y[finished] do \n" }{MPLTEXT 1 0 61 " q:=PolynomialInterpolation(X[1..d+1],y[nextv alue](),x);\n" }{MPLTEXT 1 0 34 " if q=1 or q=-1 then next fi;\n" } {MPLTEXT 1 0 32 " if polydivisorx(pp,q) then\n" }{MPLTEXT 1 0 56 " \+ return [q,op(Kroneckerfactorx(quo(pp,q,x)))] fi;\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 16 "od; [pp]; end;\n" }{MPLTEXT 1 0 2 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"pG6\"6-I\"dGF%I\"DGF%I#ppGF%I \"iGF%I\"XGF%I\"YGF%I\"vGF%I\"VGF%I\"cGF%I\"qGF%I\"yGF%F%F%C->F)-I'exp andG%*protectedGF#@$/F)\"\"!O7#F9@$/F)\"\"\"O7\"@$/F)!\"\"O7#FC>F(-I'd egreeGF66#F)>F+F@>F,F@>F*F9?(F%F>F>F%1-I%nopsGF66#F+-I&floorG6$F6I(_sy slibGF%6#,$*&#F>\"\"#F>F(F>F>C+>F--I%subsGF66$/I\"xGF%F*F)@$/F-F9O7$,& F[oF>F*FC-I#opGF66#-I1KroneckerfactorxGF%6#-I$quoGFT6%F)F`oF[o>F.-_I*n umtheoryGFTI)divisorsGF%6#F->F.-I&unionGF66$F.-I$mapGF66$f*6#F1F%6$I)o peratorGF%I&arrowGF%F%,$F1FCF%F%F%F.>F.7#-Fbo6#F.>F.-I%sortGF66$F.f*6$ I\"uGF%F-F%FipF%2-I$absGF66#Fgq-FjqF_pF%F%F%>F+7$-FboFQF*>F,7$-Fbo6#F, F.@%1F*F9>F*,&F>F>F*FC>F*,$F*FC?(F'F9F>F%1,$*&FZF>F'F>F>F(C$>F1-_I)com binatGFTI)cartprodGF%6#&F,6#;F>,&F'F>F>F>?(F%F>F>F%4&F16#I)finishedGF% C%>F0-_I-CurveFittingGFTI8PolynomialInterpolationGF%6%&F+Ffs-&F16#I*ne xtvalueGF%F%F[o@$5/F0F>/F0FC\\@$-I-polydivisorxGF%6$F)F0O7$F0-Fbo6#-Fe o6#-Fho6%F)F0F[o7#F)F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "debug(Kroneckerfactorx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I1Kronec kerfactorxG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p:=x^5-2* x^4-2*x^3+4*x^2+x-2; Kroneckerfactorx(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*$)I\"xG6\"\"\"&\"\"\"F(*&\"\"#F()F%\"\"%F(!\"\"*&F*F() F%\"\"$F(F-*&F,F()F%F*F(F(F%F(F*F-" }}{PARA 9 "" 1 "" {TEXT 208 62 "\{ --> enter Kroneckerfactorx, args = x^5-2*x^4-2*x^3+4*x^2+x-2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*$)I\"xG6\"\"\"&\"\"\"F(*&\"\"#F()F%\"\"%F(! \"\"*&F*F()F%\"\"$F(F-*&F,F()F%F*F(F(F%F(F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$ \"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&!\"#!\"\"\"\"\"\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 9 "" 1 "" {TEXT 208 54 "\{--> enter Kroneckerfactor x, args = x^4-x^3-3*x^2+x+2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I\"x G6\"\"\"%\"\"\"F(*$)F%\"\"$F(!\"\"*&F+F()F%\"\"#F(F,F%F(F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&!\"#!\" \"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7&\"\"\" !\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 9 "" 1 "" {TEXT 208 46 "\{--> enter K roneckerfactorx, args = x^3-3*x-2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(* $)I\"xG6\"\"\"$\"\"\"F(*&F'F(F%F(!\"\"\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$ \"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&!\"#!\"\"\"\"\"\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\"\"\"\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<(!\"%!\"#!\"\"\"\"\"\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(!\"%!\"#!\"\"\"\"\"\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(\"\"\"!\"\"\"\"#!\"#\"\"%!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7&\"\"\" !\"\"\"\"#!\"#7(F$F%F&F'\"\"%!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&falseG%*protectedGE\\[l#I)fi nishedGF#F$I*nextvalueGF#f*F#6%I\"pGF#I&valueGF#I\"iGF#6#IgnCopyright~ (c)~1998~Waterloo~Maple~Inc.~~All~rights~reserved.GF#F#C&@$&I\"TGF#6#. F'YQQattempt~to~call~nextvalue~on~a~finished~iteratorF#>F,7#-I$seqGF%6 $-I#opGF%6$&I(counterGF#6#F-&I'spacesGF#FB/F-;\"\"\"I*dimensionGF#>F+F H?(F#FGFGF#I%trueGF%C$>&FA6#F+,&FNFGFGFG@%2&I,nops_spacesGF#FOFNC%>FNF G>F+,&F+FGFG!\"\"@$/F+\"\"!C$>F2.FKOF,FjnF#F#6,F3F3FAFAFDFDFHFHFTFT" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&falseG%*prote ctedGE\\[l#I)finishedGF#F$I*nextvalueGF#f*F#6%I\"pGF#I&valueGF#I\"iGF# 6#IgnCopyright~(c)~1998~Waterloo~Maple~Inc.~~All~rights~reserved.GF#F# C&@$&I\"TGF#6#.F'YQQattempt~to~call~nextvalue~on~a~finished~iteratorF# >F,7#-I$seqGF%6$-I#opGF%6$&I(counterGF#6#F-&I'spacesGF#FB/F-;\"\"\"I*d imensionGF#>F+FH?(F#FGFGF#I%trueGF%C$>&FA6#F+,&FNFGFGFG@%2&I,nops_spac esGF#FOFNC%>FNFG>F+,&F+FGFG!\"\"@$/F+\"\"!C$>F2.FKOF,FjnF#F#6,F3F3FAFA FDFDFHFHFTFT" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%!\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F%F%" }}{PARA 9 "" 1 "" {TEXT 208 44 "\{ --> enter Kroneckerfactorx, args = x^2-x-2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(F%!\"\"F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$ \"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&!\"#!\"\"\"\"\"\"\"#" } }{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7 &\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7&\"\"\"!\"\"\"\"#!\"#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&fal seG%*protectedGE\\[l#I)finishedGF#F$I*nextvalueGF#f*F#6%I\"pGF#I&value GF#I\"iGF#6#IgnCopyright~(c)~1998~Waterloo~Maple~Inc.~~All~rights~rese rved.GF#F#C&@$&I\"TGF#6#.F'YQQattempt~to~call~nextvalue~on~a~finished~ iteratorF#>F,7#-I$seqGF%6$-I#opGF%6$&I(counterGF#6#F-&I'spacesGF#FB/F- ;\"\"\"I*dimensionGF#>F+FH?(F#FGFGF#I%trueGF%C$>&FA6#F+,&FNFGFGFG@%2&I ,nops_spacesGF#FOFNC%>FNFG>F+,&F+FGFG!\"\"@$/F+\"\"!C$>F2.FKOF,FjnF#F# 6,F3F3FAFAFDFDFHFHFTFT" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&falseG%*protectedGE\\[l#I)finishedGF#F$I*nextvalueG F#f*F#6%I\"pGF#I&valueGF#I\"iGF#6#IgnCopyright~(c)~1998~Waterloo~Maple ~Inc.~~All~rights~reserved.GF#F#C&@$&I\"TGF#6#.F'YQQattempt~to~call~ne xtvalue~on~a~finished~iteratorF#>F,7#-I$seqGF%6$-I#opGF%6$&I(counterGF #6#F-&I'spacesGF#FB/F-;\"\"\"I*dimensionGF#>F+FH?(F#FGFGF#I%trueGF%C$> &FA6#F+,&FNFGFGFG@%2&I,nops_spacesGF#FOFNC%>FNFG>F+,&F+FGFG!\"\"@$/F+ \"\"!C$>F2.FKOF,FjnF#F#6,F3F3FAFAFDFDFHFHFTFT" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I \"xG6\"F%!\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F% F%" }}{PARA 9 "" 1 "" {TEXT 208 40 "\{--> enter Kroneckerfactorx, args = x-2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"\"#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&!\"#!\" \"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&!\"#!\"\"\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"\"!\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7&\"\"\" !\"\"\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&falseG%*protectedGE\\[l#I)finishedGF#F$I*nextv alueGF#f*F#6%I\"pGF#I&valueGF#I\"iGF#6#IgnCopyright~(c)~1998~Waterloo~ Maple~Inc.~~All~rights~reserved.GF#F#C&@$&I\"TGF#6#.F'YQQattempt~to~ca ll~nextvalue~on~a~finished~iteratorF#>F,7#-I$seqGF%6$-I#opGF%6$&I(coun terGF#6#F-&I'spacesGF#FB/F-;\"\"\"I*dimensionGF#>F+FH?(F#FGFGF#I%trueG F%C$>&FA6#F+,&FNFGFGFG@%2&I,nops_spacesGF#FOFNC%>FNFG>F+,&F+FGFG!\"\"@ $/F+\"\"!C$>F2.FKOF,FjnF#F#6,F3F3FAFAFDFDFHFHFTFT" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "7#,&I\"xG6\"\"\"\"\"\"#!\"\"" }}{PARA 9 "" 1 "" {TEXT 208 61 "<-- exit Kroneckerfactorx (now in Kroneckerfacto rx) = [x-2]\}" }{TEXT 208 68 "\n<-- exit Kroneckerfactorx (now in Kron eckerfactorx) = [x+1, x-2]\}" }{TEXT 208 73 "\n<-- exit Kroneckerfacto rx (now in Kroneckerfactorx) = [x+1, x+1, x-2]\}" }{TEXT 208 78 "\n<-- exit Kroneckerfactorx (now in Kroneckerfactorx) = [x-1, x+1, x+1, x-2 ]\}" }{TEXT 208 76 "\n<-- exit Kroneckerfactorx (now at top level) = [ x-1, x-1, x+1, x+1, x-2]\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "7',&I\"xG6 \"\"\"\"F&!\"\"F#,&F$F&F&F&F(,&F$F&\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "undebug(Kroneckerfactorx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I1KroneckerfactorxG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Kroneckerfactorx(6*x^2+12*x+12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"#\"\"$,(*$)I\"xG6\"F#\"\"\"F**&F#F*F(F*F*F#F*" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.86. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.87. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.88. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->8.3.89. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 31 "*8.3.90. Carda no-k\303\251plet." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "p:='p'; q:='q'; solve(x^3+p*x+q=0,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"pG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"qG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&*&#\"\"\"\"\"'F&),&*& \"$3\"F&I\"qG6\"F&!\"\"*&\"#7F&),&*&F0F&)I\"pGF-\"\"$F&F&*&\"#\")F&)F, \"\"#F&F&#F&F:F&F&#F&F6F&F&*(F:F&F5F&F(F.F.,(*&#F&F0F&F(F&F.*&F5F&F(F. F&*(*&F;F&^#F&F&F&)F6F;F&,&F$F&F=F&F&F&,(F?F.FAF&*(,$FCF&F&FEF&FFF&F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r:='r'; solve(x^4+p*x^2+ q*x+r=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"rG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'RootOfG6$%*protectedGI(_syslibG6\"6#,**$)I#_ZGF$\" \"%\"\"\"F.*&I\"pGF'F.)F,\"\"#F.F.*&I\"qGF'F.F,F.F.I\"rGF'F." }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.91. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.92. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 44 "*8.3.93. Testb\305\221v\303\255t \303\251sek." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*8.3.94. T\303\251tel." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "*8.3.95. Feladat: t" }{TEXT 207 34 "estb\305\221v\303\25 5t\303\251sek." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.96. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.97. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.98. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*8.3.99. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.100. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.101. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.102. Fela dat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.10 3. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 31 " *8.3.104. Feladat: szerkeszthet" }{TEXT 207 19 "\305\221s\303\251g." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.105. Fela dat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.10 6. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 " *8.3.107. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.108. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.109. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 47 "8.3.110. V\303\251ges testek alapt\303\251tele." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "8.3.111. Wed derburn t\303\251tele." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 25 "*8.3.112. Polinomfaktoriz" }{TEXT 207 8 "\303\241" } {TEXT 207 1 "l" }{TEXT 207 8 "\303\241" }{TEXT 207 29 "s v\303\251ges \+ testek felett." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 22 "*8.3.113. Magasabb fok" }{TEXT 207 8 "\303\272" }{TEXT 207 11 " ko ngruenci" }{TEXT 207 8 "\303\241" }{TEXT 207 2 "k." }{TEXT 207 0 "" }} }{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.114. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 23 "*8.3.115. Hensel-lemma." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "*8.3.116. Me gjegyz\303\251s." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "8.3.117. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.118. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 17 "8.3.119. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.120. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 63 "8.3.121. Racion\303\241lis t\303 \266rtf\303\274ggv\303\251nyek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "r1:=(z^3-1)/(z^2-1); r1:=sim plify(r1); r2:=simplify((z^4-1)/(z^3-1));\n" }{MPLTEXT 1 0 6 "r1*r2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*$)I\"zG6\"\"\"$\"\"\"F)F)!\"\"F), &*$)F&\"\"#F)F)F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*$)I\"zG6\" \"\"#\"\"\"F)F&F)F)F)F),&F&F)F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,**$)I\"zG6\"\"\"$\"\"\"F)*$)F&\"\"#F)F)F&F)F)F)F),(F*F)F&F)F)F)! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"zG6\"\"\"\"F&F&!\"\",**$) F$\"\"$F&F&*$)F$\"\"#F&F&F$F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8. 3.122. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 56 "8.3.123. Parci\303\241lis t\303\266rtekre bont\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f :=1/(x^4-x^2); convert(f,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " *$,&*$)I\"xG6\"\"\"%\"\"\"F)*$)F&\"\"#F)!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#!\"\"F(*&#\"\"\"F'F+,&F%F+F+F(F(F+*&F*F +,&F%F+F+F+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}} }{SECT 0 {PARA 5 "" 0 "" {TEXT 207 36 "8.3.124. K\303\266vetkezm\303\2 51ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:=36/(x^5-2*x^4-2*x^3+4*x^2+x-2); convert(f,parfrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"#O\"\"\",.*$)I\"xG6\"\"\"&F %F%*&\"\"#F%)F)\"\"%F%!\"\"*&F-F%)F)\"\"$F%F0*&F/F%)F)F-F%F%F)F%F-F0F0 F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"$\"\"\"),&I\"xG6\"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 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 36 "8.3.125. K\303\266vetkezm\303\251ny." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f:=(x^5+1)/(x^4-x^2); convert(f,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*$)I\"xG6\"\"\"&\"\"\"F)F)F)F),&*$)F&\"\"%F)F)*$)F&\" \"#F)!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(I\"xG6\"\"\"\"*$)F#\" \"#!\"\"F)*$,&F#F%F%F)F)F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 27 "8.3.126. Megjegyz\30 3\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "b:='b'; f:=x/(x-b)^2; convert(f,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"bG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I \"xG6\"\"\"\"),&F#F%I\"bGF$!\"\"\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I\"bG6\"\"\"\"),&I\"xGF%F&F$!\"\"\"\"#F*F&*$F(F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f:=(4*x^3-6*x^2-2)/(x^4-2*x^3-2*x+4 ); convert(f,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(*&\"\"% \"\"\")I\"xG6\"\"\"$F&F&*&\"\"'F&)F(\"\"#F&!\"\"F.F/F&,**$)F(F%F&F&*&F .F&F'F&F/*&F.F&F(F&F/F%F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(\"\"$ \"\"\")I\"xG6\"\"\"#F%,&*$)F'F$F%F%F)!\"\"F-F%*$,&F'F%F)F-F-F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "convert(f,parfrac,x,2^(1/3)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(**,&\"\"\"!\"\"*$)\"\"##F)\"\"$F% F%F%)F)#F%F+F%,&I\"xG6\"F%*$F,F%F&F&,&F)F&F1F%F&F&**\"\"'F%,&F/F%F)F&F &,(\"\"%F%*&F)F%F,F%F%F'F%F&F2F&F&*&,&*&F)F%F/F%F&F1F&F%,(*$)F/F)F%F&* &F/F%F,F%F&F'F&F&F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 19 "->8.3.127. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 58 "8.3.128. T\303\266b bhat\303\241rozatlan\303\272 polinomok." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "p:=3*x^3*y^2+4*x^3*y+ 7*x^2*y^2*z+9; indets(p); coeff(p,x^3); coeff(p,y^2);\n" }{MPLTEXT 1 0 42 "degree(p,x); degree(p,\{x,z\}); degree(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**(\"\"$\"\"\")I\"xG6\"F$F%)I\"yGF(\"\"#F%F%*(\"\"%F%F&F %F*F%F%**\"\"(F%)F'F+F%F)F%I\"zGF(F%F%\"\"*F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%I\"xG6\"I\"zGF$I\"yGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"$\"\"\")I\"yG6\"\"\"#F%F%*&\"\"%F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"$\"\"\")I\"xG6\"F$F%F%*(\"\"(F%)F'\"\"#F%I\"zGF(F %F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 23 "*8.3.129. Multiindexek." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 46 "*8.3.130. Form\303\241lis hatv\303\241nysorok." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "8.3.131. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 27 "8.3. 132. Megjegyz\303\251s." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 33 "*8.3.133. Szimmetrikus polinomok." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 51 " *8.3.134. Szimmetrikus polinomok alapt\303\251tele." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 33 "*8.3.135. Newton k\303\251pletei." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->8. 3.136. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.137. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.138. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 18 "*8.3.139. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.140. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "*8.3.141. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 12 "8.3.142. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 20 "bbi feladatok megold" }{TEXT 207 15 " \303\241sokkal." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 12 "8.3.143. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 14 "bbi feladatok ." }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" } }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 24 "9. K\303\263dol\303\241s" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 16 "1 0. Algoritmusok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }