JSFH

TP 3

Exercice 1

restart;

with(numtheory):

racines:=proc(p::prime,m::posint)

local L;

L:=convert(divisors(p^m-1),list)[2..-1];

return select(n->order(p,n)=m,L)

end proc:

p:=5;

IiIm

m:=6;

IiIn

p^m;

IiZEYyI=

N:=racines(p,m);

N0ciIigiIioiIzkiIz0iI0AiI0ciI08iI1UiI2MiI2oiI3MiIyUpIiMkKiIkRSIiJG8iIiQnPSIkPCMiJFsjIiRfIyIkeiMiJHMkIiRNJSIkLyYiJGUmIiReJyIkVygiJG8pIiU7NiIlLTgiJU88IiVgPiIlS0EiJS9FIiUxUiIlM18iJTd5IiZDYyI=

F:=map(n->iquo(phi(n),m),N);

N0ciIiJGI0YjRiMiIiNGJEYkRiQiIiUiIidGJUYlIiM1RiYiIilGJyIjSSIjPyIjN0YpRipGKSIjQ0YpIiNnIiNTRi1GLUYtIiQ/IiIkIT1GL0YvRjAiJFMjIiRnJCIkPyg=

add(n,n in F); # nombre de corps finis distincts (mais tous isomorphes entre eux) de cardinal 15625

IiUhZSM=

Exercice 2

n:=N[19];

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F[19];

IiM3

Phi:=cyclotomic(n,X);

LDQqJEkiWEc2IiIjcyIiIiokRiQiI21GJyokRiQiI2EhIiIqJEYkIiNbRiwqJEYkIiNPRicqJEYkIiNDRiwqJEYkIiM9RiwqJEYkIiInRidGJ0Yn

d:=iquo(phi(n),m); # modulo p F se factorise en 12 facteurs tous de degr\303\251 6

IiM3

FF:=Factors(Phi) mod p;

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

LFF:=map(F->F[1],FF[2]);

Ny4sMCokSSJYRzYiIiInIiIiKiRGJSIiJiIiJSokRiVGKyIiJCokRiVGLUYrKiRGJSIiI0YwRiVGK0YrRigsLEYkRihGLEYrRi5GK0YvRihGK0YoLDBGJEYoRilGKEYsRjBGLkYrRi9GLUYlRihGK0YoLCxGJEYoRilGKEYuRitGJUYoRitGKCwuRiRGKEYpRihGLEYwRi9GLUYlRihGK0YoLC5GJEYoRilGK0YsRjBGL0YtRiVGK0YrRigsLEYkRihGLEYrRi5GKEYvRihGK0YoLC5GJEYoRilGMEYsRi1GL0YwRiVGMEYrRigsLEYkRihGKUYrRi5GKEYlRitGK0YoLC5GJEYoRilGLUYsRi1GL0YwRiVGLUYrRigsMEYkRihGKUYrRixGMEYuRihGL0YtRiVGK0YrRigsMEYkRihGKUYoRixGLUYuRihGL0YwRiVGKEYrRig=

P:=LFF[4]; # on choisit l'un des facteurs irr\303\251ductibles de F mod p

LCwqJEkiWEc2IiIiJyIiIiokRiQiIiZGJyokRiQiIiQiIiVGJEYnRixGJw==

alias(x=RootOf(P,X) mod p); # d\303\251finit l'\303\251l\303\251ment primitif x de K=F_p[x]$

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Power(x,6) mod p;

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Power(x,n) mod p;Power(x,n/2) mod p; # x est bien d'ordre 256

IiIi

IiIl

d:= trunc(sqrt(n));

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n_sur_d:=iquo(n,d);

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U:=[seq(Power(x,r) mod p, r=0..d-1)];

NzEiIiJJInhHNiIqJEYkIiIjKiRGJCIiJCokRiQiIiUqJEYkIiImLCpGLEYrRihGI0YkRitGI0YjLC5GLEYjRipGI0YoRitGJkYrRiRGJ0YrRiMsKkYqRitGJkYnRiRGKUYjRiMsKkYsRitGKEYnRiZGKUYkRiMsLkYsRiNGKkYnRihGJ0YmRiNGJEYjRitGIywuRixGI0YqRidGKEYnRiZGI0YkRilGI0YjLCxGLEYjRipGJ0YoRidGJkYpRiNGIywqRixGI0YqRidGKEYrRiNGIywqRixGI0YqRitGKEYjRiNGIw==

y:=Power(x,n-d) mod p;

LCwqJEkieEc2IiIiJiIiJCokRiQiIiVGJyokRiRGJ0YnKiRGJCIiI0YsRiwiIiI=

V:=[seq(Power(y,q) mod p, q=0..n_sur_d)];

NzMiIiIsLCokSSJ4RzYiIiImIiIkKiRGJiIiJUYpKiRGJkYpRikqJEYmIiIjRi5GLkYjLCxGJUYrRipGLkYsRi5GJkYuRilGIywoRipGK0YsRilGK0YjLCRGLEYpLCxGJUYjRixGI0YtRitGJkYjRitGIywqRixGK0YtRilGJkYuRi5GIywsRiVGKUYqRi5GLEYrRi1GKUYuRiMsKkYlRiNGLEYrRiZGI0YrRiMsKkYlRitGLUYjRiZGI0YjRiMsLEYlRi5GKkYjRixGKUYmRilGLkYjLCxGJUYpRipGLkYsRi5GLUYuRiZGLiwqRiVGKUYsRitGLUYjRiZGLiwuRiVGKUYqRiNGLEYpRi1GK0YmRiNGLkYjLCxGJUYrRipGI0YsRi5GLUYrRi5GIywsRiVGI0YqRilGLUYuRiZGKUYrRiMsLEYlRiNGKkYuRixGLkYtRilGI0Yj

z1:=x^5+2*x^2+x+3;

LCoqJEkieEc2IiIiJiIiIiokRiQiIiNGKUYkRiciIiRGJw==

for v in V do

zz:=evala(z1*v) mod p;

if member(zz,U,'r') then

member(v,V,'q');q:=q-1;r:=r-1;

break;

end if;

end do:

k:=q*d+r:

q,r,k,evalb(z1=Power(x,k) mod p);

Error, (in mod/Power) 2nd argument (exponent) must be a non-negative integer

z2:=3*x^4+3*x^3+x^2+2*x+4;

LCwqJEkieEc2IiIiJSIiJCokRiRGJ0YnKiRGJCIiIyIiIkYkRipGJkYr

unassign('q','r');