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-1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" 
{TEXT -1 80 "Warning, the protected names norm and trace have been red
efined and unprotected\n" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Berle
kamp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "berlekamp:=proc(u,x
, p)\nlocal n,k,rk,Q,i,lst;\n\nn:=degree(u,x);\nQ := matrix(n,n);\nfor
 k from 0 to n - 1 do\n  rk := Rem(x^(k*p),u,x) mod p ;\n  for i from \+
1 to n do \n  Q[i,k+1] := coeff(rk,x,i-1)\n  od od;\nlst := convert(Nu
llspace(charmat(Q,1)) mod p, list);\nlst := [seq(add(lst[j][i]*x^(i-1)
, i=1..degree(f,x)),j=1..nops(lst))];\nend;\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%*berlekampGR6%%\"uG%\"xG%\"pG6(%\"nG%\"kG%#rkG%\"QG%
\"iG%$lstG6\"F1C'>8$-%'degreeG6$9$9%>8'-%'matrixG6$F4F4?(8%\"\"!\"\"\"
,&F4FBFB!\"\"%%trueGC$>8&-%$modG6$-%$RemG6%)F9*&F@FB9&FBF8F9FQ?(8(FBFB
F4FE>&F;6$FS,&F@FBFBFB-%&coeffG6%FHF9,&FSFBFBFD>8)-%(convertG6$-FJ6$-%
*NullspaceG6#-%(charmatG6$F;FBFQ%%listG>Fgn7#-%$seqG6$-%$addG6$*&&&Fgn
6#%\"jG6#FSFB)F9FenFB/FS;FB-F66$%\"fGF9/F`p;FB-%%nopsG6#FgnF1F1F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "scinde := proc(u,v,x,p)\nlo
cal lst,s;\nlst := [seq(Gcd(u,v-s) mod p, s=0..p-1)];\nop(select(t -> \+
degree(t,x) > 0, lst))\nend;          " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'scindeGR6&%\"uG%\"vG%\"xG%\"pG6$%$lstG%\"sG6\"F.C$>8$7#-%$seq
G6$-%$modG6$-%$GcdG6$9$,&9%\"\"\"8%!\"\"9'/F@;\"\"!,&FBF?F?FA-%#opG6#-
%'selectG6$R6#%\"tGF.6$%)operatorG%&arrowGF.2FE-%'degreeG6$F<T$F.F.6$F
)9&F1F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Passons maintenant
 a la factorisation :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "berlfact \+
:= proc(f,x,p)\nlocal r,base,res,j,v;\nbase := berlekamp(f,x,p);\nr:=n
ops(base);\nres := [f];\nj := 0;\nwhile(nops(res) < r) do\n  j := j+1;
\n  v := base[j];\n  if (degree(v,x) > 0) then res := map(scinde,res,v
,x,p) fi;\n  od;\nres \nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)ber
lfactGR6%%\"fG%\"xG%\"pG6'%\"rG%%baseG%$resG%\"jG%\"vG6\"F0C(>8%-%*ber
lekampG6%9$9%9&>8$-%%nopsG6#F3>8&7#F7>8'\"\"!?(F0\"\"\"FFF02-F=6#F@F;C
%>FC,&FCFFFFFF>8(&F36#FC@$2FD-%'degreeG6$FNF8>F@-%$mapG6'%'scindeGF@FN
F8F9F@F0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x^60 -
 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$)%\"xG\"#g\"\"\"F*F*!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "convert(berlfact(f,
x,7),`*`);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*L,&%\"xG\"\"\"F&F&F&,&F
%F&\"\"'F&F&,,*$)F%\"\"%F&F&*&F(F&)F%\"\"$F&F&*$)F%\"\"#F&F&*&F(F&F%F&
F&F&F&F&,,F*F&*$F.F&F&F0F&F%F&F&F&F&,&F0F&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&F&*&F,F&F1F&F&*&F/F&F%F&F&F&
F&F&,&F0F&F2F&F&,,F*F&*&F2F&F.F&F&F0F&*&F/F&F%F&F&F,F&F&,,F*F&*&\"\"&F
&F.F&F&F0F&*&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
&*&F2F&F1F&F&*&F(F&F%F&F&F,F&F&,,F*F&*&F,F&F.F&F&*&F2F&F1F&F&F%F&F,F&F
&,&F%F&FEF&F&,&F%F&F2F&F&,,F*F&*&F2F&F.F&F&*&F,F&F1F&F&F%F&F2F&F&,,F*F
&*&FEF&F.F&F&*&F,F&F1F&F&*&F(F&F%F&F&F2F&F&,&F0F&F,F&F&,,F*F&F5F&*&F2F
&F1F&F&*&F/F&F%F&F&F2F&F&,,F*F&*&F(F&F.F&F&*&F2F&F1F&F&*&F,F&F%F&F&F2F
&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Factor(f) mod 7;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*L,&%\"xG\"\"\"F&F&F&,&F%F&\"\"'F&F&,,
*$)F%\"\"%F&F&*&F(F&)F%\"\"$F&F&*$)F%\"\"#F&F&*&F(F&F%F&F&F&F&F&,,F*F&
*$F.F&F&F0F&F%F&F&F&F&,&F0F&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&F&*&F,F&F1F&F&*&F/F&F%F&F&F&F&F&,&F0F&F2F&
F&,,F*F&*&F2F&F.F&F&F0F&*&F/F&F%F&F&F,F&F&,,F*F&*&\"\"&F&F.F&F&F0F&*&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&*&F2F&F1F&F&*
&F(F&F%F&F&F,F&F&,,F*F&*&F,F&F.F&F&*&F2F&F1F&F&F%F&F,F&F&,&F%F&FEF&F&,
&F%F&F2F&F&,,F*F&*&F2F&F.F&F&*&F,F&F1F&F&F%F&F2F&F&,,F*F&*&FEF&F.F&F&*
&F,F&F1F&F&*&F(F&F%F&F&F2F&F&,&F0F&F,F&F&,,F*F&F5F&*&F2F&F1F&F&*&F/F&F
%F&F&F2F&F&,,F*F&*&F(F&F.F&F&*&F2F&F1F&F&*&F,F&F%F&F&F2F&F&" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Result
ants et polynome annulateur." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
77 "polysom := proc(P,Q,x)\nlocal y ;\nsubs(y=x,resultant(subs(x=y-x,Q
),P,x));\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(polysomGR6%%\"PG%
\"QG%\"xG6#%\"yG6\"F,-%%subsG6$/8$9&-%*resultantG6%-F.6$/F2,&F1\"\"\"F
2!\"\"9%9$F2F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "polys
om(x^2-2,x^2-5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#
\"\"\"!#9*$)F&\"\"%F(F(\"\"*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 113 "polyprod := proc(P,Q,x)\nlocal y,Q1;\nQ1 := expand(x^(degree(
Q,x))*subs(x=y/x,Q));\nsubs(y=x,resultant(Q1,P,x))\nend;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%)polyprodGR6%%\"PG%\"QG%\"xG6$%\"yG%#Q1G6\"F-C
$>8%-%'expandG6#*&)9&-%'degreeG6$9%F6\"\"\"-%%subsG6$/F6*&8$F;F6!\"\"F
:F;-F=6$/FAF6-%*resultantG6%F09$F6F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "polyrad := proc(P,n,x)\nsubs(x=x^n,P)\nend;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(polyradGR6%%\"PG%\"nG%\"xG6\"F*F*-%%subsG
6$/9&)F/9%9$F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 
112 "La procedure choose recoit une expression a et un polynome annula
teur A  de  cette expression. Elle factorise  A" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "et rend le facteur irreductible de A qui admet a pour rac
ine." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "nettoie := proc(a,A,x)\n l
ocal lst;\n lst := op(2,factors(A));\n lst := map(t -> op(1,t),lst);\n
 op(select(t -> abs(evalf(subs(x=a,t))) < 0.01, lst))\nend;\n" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(nettoieGR6%%\"aG%\"AG%\"xG6#%$lstG6
\"F,C%>8$-%#opG6$\"\"#-%(factorsG6#9%>F/-%$mapG6$R6#%\"tGF,6$%)operato
rG%&arrowGF,-F16$\"\"\"9$F,F,F,F/-F16#-%'selectG6$RF=F,F?F,2-%$absG6#-
%&evalfG6#-%%subsG6$/T$T&FE$FD!\"#F,F,6&F)9&F'FEF/F,F,F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 563 "polyannulateur := proc(e,x)\nlocal
 exp,A;\nif type(e,integer) then A := x-e\nelif type(e,symbol) then A \+
:= x -e  \nelif type(e,`+`) then A := polysom(polyannulateur(op(1,e),x
),polyannulateur(e-op(1,e),x),x)\nelif type(e,`*`) then if op(1,e) = -
1 then A := subs(x=-x,polyannulateur(op(2,e),x))\n                 els
e A := polyprod(polyannulateur(op(1,e),x),polyannulateur(op(2,e),x),x)
                  fi\nelif type(e,`^`) then exp := 1/op(2,e); if not t
ype(exp,integer) then RETURN(FAIL)\n    else A := polyrad(polyannulate
ur(op(1,e),x),exp,x) fi\nfi;\nnettoie(e,A,x)\nend;" }}{PAGEBK }{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%/polyannul
ateurGR6$%\"eG%\"xG6$%$expG%\"AG6\"F,C$@,-%%typeG6$9$%(integerG>8%,&9%
\"\"\"F2!\"\"-F06$F2%'symbolG>F5F6-F06$F2%\"+G>F5-%(polysomG6%-F$6$-%#
opG6$F8F2F7-F$6$,&F2F8FGF9F7F7-F06$F2%\"*G@%/FGF9>F5-%%subsG6$/F7,$F7F
9-F$6$-FH6$\"\"#F2F7>F5-%)polyprodG6%FEFXF7-F06$F2%\"^GC$>8$*&F8F8FZF9
@%4-F06$F`oF3-%'RETURNG6#%%FAILG>F5-%(polyradG6%FEF`oF7-%(nettoieG6%F2
F5F7F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a := sqrt(5)+
sqrt(22+2*sqrt(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&*$-%%sq
rtG6#\"\"&\"\"\"F+*$-F(6#,&\"#AF+*&\"\"#F+F'F+F+F+F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "A := polyannulateur(a,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG,**$)%\"xG\"\"%\"\"\"F**&\"#aF*)F(\"\"#F*!\"
\"*&\"#SF*F(F*F/\"$p#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
evalf(subs(x=a,A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"$!\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "f := (11 + 2*(29)^(1/2))^(1/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG*$-%%sqrtG6#,&\"#6\"\"\"*&\"\"#F+-F'6#\"#HF+F+F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "g := (16 - 2*29^(1/2) +
 2*(55 - 10*(29)^(1/2))^(1/2))^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"gG*$-%%sqrtG6#,(\"#;\"\"\"*&\"\"#F+-F'6#\"#HF+!\"\"*&F-F+-F'6#,
&\"#bF+*&\"#5F+F.F+F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "b := f+g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,&*$-%%sqrtG6#,&
\"#6\"\"\"*&\"\"#F,-F(6#\"#HF,F,F,F,*$-F(6#,(\"#;F,*&F.F,F/F,!\"\"*&F.
F,-F(6#,&\"#bF,*&\"#5F,F/F,F8F,F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "B := polyannulateur(b,x);" }}{PAGEBK }{PARA 256 "" 0 
"" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,**$)%\"xG\"
\"%\"\"\"F**&\"#aF*)F(\"\"#F*!\"\"*&\"#SF*F(F*F/\"$p#F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(subs(x=b,B));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"#:!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 4 "A-B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "fsolve(B,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6&$!5j!*GB\"f<(pAk!#>$!5#y&yg*e)*R!4HF%$\"5N^\\F\"4Ah0&>F%$\"55(zl
&*3%f<\"Q(F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(a) ; \+
evalf(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"55(zl&*3%f<\"Q(!#>" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"56(zl&*3%f<\"Q(!#>" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }