Factorisation de polynômes modulo p

p
p f n
Z/pZ
f
f f Xp
f f Xp
f Af
Z/pZ Z/pZ[X]/(f)f
x7→ xp−x Af(¯ 1, X, . . . , Xn−1)U
AfU∈Z/pZ[X]
f
f f
f
f
f(X) = X6+X3+ 1
fFp
p≡1 (mod 9)
3< p 61000
Fp[X]p
3N N
Z[X]
Qp
36p6N
X4+ 1 Q

100
f=X4+8X+12 ∈Z[X]Q
X4+ 1
p f X4+ 1
p
f∈Fp[X]
f∈Fp[X]
p(g1, . . . , gs)Qigi=f gi
i
g∈
Fp[X] 2 2 d
Φn
p-nFp[X]
p(Z/nZ)×
f∈Fp[X]