{VERSION 6 0 "SGI MIPS UNIX" "6.0" } {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 "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 10 "Computing " }{XPPEDIT 18 0 "sqrt(I);" "6#-%%sqrtG6#%\"IG" }{TEXT -1 25 " the radical of an idea l." }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F := \+ [x*y^2+2*y^2,x^4-2*x^2+1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7$ ,&*&%\"xG\"\"\")%\"yG\"\"#F)F)*&F,F)F*F)F),(*$)F(\"\"%F)F)*&F,F))F(F,F )!\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "G := Groebner [Basis](F,plex(y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7$,(*$) %\"xG\"\"%\"\"\"F+*&\"\"#F+)F)F-F+!\"\"F+F+*$)%\"yGF-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&),&%\"xG\"\"\"F(!\"\"\"\"#F(),&F'F(F(F(F*F(*$)%\"yG F*F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "From the factorization, t he variety " }{TEXT 259 1 "V" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "(x-1)^2* (x+1)^2;" "6#*&,&%\"xG\"\"\"F&!\"\"\"\"#,&F%F&F&F&F(" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "y^2;" "6#*$%\"yG\"\"#" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "V(x^2-1,y);" "6#-%\"VG6$,&*$%\"xG\"\"#\"\"\"F*!\"\"%\"yG" } {TEXT -1 1 " " }{TEXT -1 8 ". Thus " }{XPPEDIT 18 0 "sqrt(I);" "6#-%% sqrtG6#%\"IG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "`<,>`(x^2-1,y);" "6#- %$<,>G6$,&*$%\"xG\"\"#\"\"\"F*!\"\"%\"yG" }{TEXT -1 1 " " }}}{EXCHG } {EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F := [x^2-2,y^2+2*x*y+2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"F G7$,&*$)%\"xG\"\"#\"\"\"F+F*!\"\",(*$)%\"yGF*F+F+*(F*F+F0F+F)F+F+F*F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "G := Groebner[Basis](F, plex(y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7$,&*$)%\"xG\"\"# \"\"\"F+F*!\"\",(*$)%\"yGF*F+F+*(F*F+F0F+F)F+F+F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&*$)%\"xG\"\"#\"\"\"F)F(!\"\",(*$)%\"yGF(F)F)*(F(F)F.F)F'F)F) F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "We need to factor G[2] ov er the field " }{TEXT 257 1 "Q" }{TEXT -1 4 "[z]/" }{XPPEDIT 18 0 "`<, >`(x^2-2);" "6#-%$<,>G6#,&*$)%\"xG\"\"#\"\"\"F+F*!\"\"" }{TEXT -1 18 " . \nMaple uses is " }{TEXT 258 1 "Q" }{TEXT -1 1 "(" }{XPPEDIT 18 0 " alpha;" "6#%&alphaG" }{TEXT -1 8 ") where " }{XPPEDIT 18 0 "alpha;" "6 #%&alphaG" }{TEXT -1 29 " is a root of the polynomial " }{XPPEDIT 18 0 "x^2-2;" "6#,&*$)%\"xG\"\"#\"\"\"F(F'!\"\"" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alias(alpha=RootOf(x^2-2)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&alphaG" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g := subs( x=alpha, G[2] );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"gG,(*$)%\"yG\"\"#\"\"\"F**(F)F*F(F*%&alphaGF *F*F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "'g[red]' = facto r(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"gG6#%$redG*$),&%\"yG\"\" \"%&alphaGF,\"\"#F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Now we al so know how to compute the \"square-free part\" of a polynomial in ove r a field by using GCDs." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gcd(g,diff(g,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"yG\"\"\"%&a lphaGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'g[red]' = quo(g ,gcd(g,diff(g,y)),y);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"gG6#%$ redG,&%\"yG\"\"\"%&alphaGF*" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "sqrt(I);" "6#-%%sqrtG6#% \"IG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`<,>`(x^2-2,y+x);" "6#-%$<,>G6 $,&*$)%\"xG\"\"#\"\"\"F+F*!\"\"6#,&%\"yGF+F)F+" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 " Now using the PolynomialIdeals pa ckage " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(PolynomialI deals):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "First, I want to be ab le to use the letter \"I\" for an ideal. But Maple uses it for " } {XPPEDIT 18 0 "sqrt(-1);" "6#-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 23 " . So I turn this off." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "in terface(imaginaryunit=_i):" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "I := ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6$,(*$)%\"yG\"\"#\"\"\"F-*(F,F-F+F-%\"xGF- F-F,F-,&*$)F/F,F-F-F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Groebner[Basis](I,plex(y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 $,&*$)%\"xG\"\"#\"\"\"F)F(!\"\",(*$)%\"yGF(F)F)*(F(F)F.F)F'F)F)F(F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "J := Radical(I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$<,>G6%,(*$)%\"yG\"\"#\"\"\"F-*(F,F- F+F-%\"xGF-F-F,F-,&*$)F/F,F-F-F,!\"\",&*&F+F-F/F-F-F1F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Groebner[Basis](J,plex(y,x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&*$)%\"xG\"\"#\"\"\"F)F(!\"\",&%\"y GF)F'F)" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "13 0 0" 34 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }