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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := factor(x^4 - 2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 ",&*$I\"xG6\"\"\"%\"\"\"!\"#F'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "alias(y1 = RootOf(f,x));" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "I#y1G6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f
acs_y1:=evala(Factor(f,y1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,&*$I
\"xG6\"\"\"#\"\"\"*$I#y1GF&F'F(F(,&F%F(F*F(F(,&F%F(F*!\"\"F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f1 := op(1,facs_y1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",&*$I\"xG6\"\"\"#\"\"\"*$I#y1GF%F&F'" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "alias(y2 = RootOf(f1,x));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$I#y1G6\"I#y2GF$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 39 "facs_y1_y2 := evala(Factor(f,\{y2,y1\}));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "**,&I\"xG6\"\"\"\"I#y2GF%F&F&,&F$F&F'!
\"\"F&,&F$F&I#y1GF%F&F&,&F$F&F+F)F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "facs_z := subs(\{y1=z1,y2=z2\},facs_y1_y2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "**,&I\"xG6\"\"\"\"I#z2GF%F&F&,&F$F&F'!\"\"F&,&
F$F&I#z1GF%F&F&,&F$F&F+F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "f1_z := subs(\{y1=z1,x=z2\},f1);f_z := subs(x=z1,f);" }}{PARA 11 "
" 1 "" {XPPMATH 20 ",&*$I#z2G6\"\"\"#\"\"\"*$I#z1GF%F&F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 ",&*$I#z1G6\"\"\"%\"\"\"!\"#F'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "F := expand(facs_z);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",**$I\"xG6\"\"\"%\"\"\"*&F$\"\"#I#z1GF%F)!\"\"*&I#z2GF%F)
F$F)F+*&F-F)F*F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "expan
d(rem(F,f1_z,z2));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$I\"xG6\"\"\"%
\"\"\"*$I#z1GF%F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "r
em(%,f_z,z1);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$I\"xG6\"\"\"%\"\"\"
!\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evala(Factor(f1,
\{y2,y1\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"xG6\"\"\"\"I#y2GF
%F&F&,&F$F&F'!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for
 i from 1 to 4 do ry||i := -subs(x=0,op(i,facs_y1_y2)); od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 ",$I#y2G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "I#y2G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#y1G6\"!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I#y1G6\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "# check a root by substituting back" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "for i from 1 to 4 do evala(subs(x=ry||i, \+
f)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "# Now we wish to identify an automorphism which maps y1 -> ry2\n" 
}{MPLTEXT 1 0 39 "# First, find the possible images of y2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map_y1 := ry2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I#y2G6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ma
p_f1 := evala(subs(y1 = map_y1, f1));" }}{PARA 11 "" 1 "" {XPPMATH 20 
",&*$I\"xG6\"\"\"#\"\"\"*$I#y1GF%F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "# map_f1 is a factor of f over Q(y1,y2)\n" }{MPLTEXT 
1 0 61 "evala(Factor(map_f1, \{y2,y1\})); # could have plugged in root
s" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"xG6\"\"\"\"I#y1GF%!\"\"F&,&F
$F&F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Possible ima
ges for y2 are ry3 and ry4\n" }{MPLTEXT 1 0 14 "map_y2 := ry3;" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",$I#y1G6\"!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "evala(subs(x=map_y2,map_f1));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "# C
ompute the permutation of the roots by mapping them\n" }{MPLTEXT 1 0 
21 "for i from 1 to 4 do\n" }{MPLTEXT 1 0 59 "    map_ry||i := evala(s
ubs(\{y1=map_y1,y2=map_y2\},ry||i));\n" }{MPLTEXT 1 0 16 "    # print(
i);\n" }{MPLTEXT 1 0 28 "    print(sort(map_ry||i));\n" }{MPLTEXT 1 0 
34 "    # print(\"Check if root of f\",\n" }{MPLTEXT 1 0 42 "         \+
   #evala(subs(x=map_ry||i,f)));\n" }{MPLTEXT 1 0 25 "    for j from 1
 to 4 do\n" }{MPLTEXT 1 0 70 "        if eval(map_ry||i - ry||j) = 0 t
hen print(cat(i,\"->\",j)); fi;\n" }{MPLTEXT 1 0 8 "    od;\n" }
{MPLTEXT 1 0 7 "end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#y1G6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I%1->4G6\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 ",$I#y1G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%2->3G6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",$I#y2G6\"!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "I%3->1G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#y2G6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I%4->2G6\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "# Compute the norm of a random element" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "u:=  RandomTools['Generate'](polyno
m(integer(range=-10..10),\{z1,z2\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 
",H!\"*\"\"\"I#z1G6\"\"#5I#z2GF&!\"#*&F%F$F(F$F#*$F%\"\"#\"\")*$F(F,!
\"(*&F(F,F%F$!\"&*&F(F$F%F,F/*$F%\"\"$F/*$F(F4!\"\"*&F(F,F%F,!#7*&F%F$
F(F4\"\"&*&F(F$F%F4F#*$F(\"\"%F'*&F%F,F(F4F=*&F%F$F(F=!\")*&F(F$F%F=!
\"'*$F%F:F4*$F(F:!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "no
rm_u_y2:=evala(expand(subs(\{z1=y1,z2=y2\},u) * subs(\{z1=y1,z2=-y2\},
u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*\"%TL\"\"\"I#y1G6\"\"%w<*$F%
\"\"#\"%MD*$F%\"\"$\"$O(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 
"norm_u_y2_y1:=1: for i from 1 to 4 do \n" }{MPLTEXT 1 0 94 "         \+
           norm_u_y2_y1 := evala(expand(norm_u_y2_y1 * subs(y1=ry||i, \+
norm_u_y2))); \n" }{MPLTEXT 1 0 20 "                 od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "norm_u_y2_y1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"/,;;iKBI" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "
ifactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I!G6\"6#\"\"$\"\"#-F$6
#\".*G^8DfL\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "norm_u
x_y2:=evala(expand((x - subs(\{z1=y1,z2=y2\},u)) * (x - subs(\{z1=y1,z
2=-y2\},u))));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2\"%TL\"\"\"I#y1G6\"
\"%w<*$F%\"\"#\"%MD*$F%\"\"$\"$O(*&I\"xGF&F$F%F)!#IF/!#q*&F%F,F/F$\"\"
%*$F/F)F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evala(Factor(n
orm_ux_y2,\{y2,y1\}));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "*&,2I\"xG6\"\"\"\"*$I#y1GF%\"\"#!#:!#NF&*$F(\"
\"$F)*&F(F&I#y2GF%F&!\"*F/!#I*&F/F&F(F)!\"'*&F/F&F(F-!#9F&,2F+F&F$F&F'
F*F/\"#IF,F)F.\"\"*F2\"\"'F4\"#9F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "norm_ux_y2_y1:=1: for i from 1 to 4 do \n" }{MPLTEXT 
1 0 64 "                    norm_ux_y2_y1 := evala(expand(norm_ux_y2_y
1\n" }{MPLTEXT 1 0 83 "                                               \+
   * subs(y1=ry||i, norm_ux_y2))); \n" }{MPLTEXT 1 0 20 "             \+
    od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "norm_ux_y2_y1;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ",4*$I\"xG6\"\"\")\"\"\"*$F$\"\"(!$!G*$
F$\"\"'\"&k\"R*$F$\"\"&!(#zIJ*$F$\"\"%\"*q'4S8*$F$\"\"$!++5+\"o#*$F$\"
\"#\",?='3%4$F$!.[!\\cMf7\"/,;;iKBIF'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",4*$I\"xG
6\"\"\")\"\"\"*$F$\"\"(!$!G*$F$\"\"'\"&k\"R*$F$\"\"&!(#zIJ*$F$\"\"%\"*
q'4S8*$F$\"\"$!++5+\"o#*$F$\"\"#\",?='3%4$F$!.[!\\cMf7\"/,;;iKBIF'" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evala(subs(x=subs(\{z1=y1,z
2=y2\},u),norm_ux_y2_y1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# Compute the norm of an ele
ment" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u2:=  z1 + 2*z2;" }
}{PARA 11 "" 1 "" {XPPMATH 20 ",&I#z1G6\"\"\"\"I#z2GF$\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "norm_u2_y2:=evala(expand(sub
s(\{z1=y1,z2=y2\},u2) * subs(\{z1=y1,z2=-y2\},u2)));" }}{PARA 11 "" 1 
"" {XPPMATH 20 ",$*$I#y1G6\"\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "norm_u2_y2_y1:=1: for i from 1 to 4 do \n" }{MPLTEXT 
1 0 97 "                    norm_u2_y2_y1 := evala(expand(norm_u2_y2_y
1 * subs(y1=ry||i, norm_u2_y2))); \n" }{MPLTEXT 1 0 20 "              \+
   od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "norm_u2_y2_y1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "\"%+D" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "ifactor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I!G6
\"6#\"\"#F'-F$6#\"\"&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
89 "norm_u2x_y2:=evala(expand((x - subs(\{z1=y1,z2=y2\},u2)) * (x - su
bs(\{z1=y1,z2=-y2\},u2))));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$I#y1G
6\"\"\"#\"\"&*&I\"xGF%\"\"\"F$F*!\"#*$F)F&F*" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 35 "evala(Factor(norm_u2x_y2,\{y2,y1\}));" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(I\"xG6
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