Racines de polynômes modulo p

p
f∈Z[X]n
p f ∈Fp[X]Fp
Oa0=f(0) 6= 0
f=anXn+· · · =a1X+a0
f(a0X) = an
0Xn+· · · +a0a1X+a0
=a0(an−1
0Xn+· · · +a1X+ 1)
f(0) = 1
M g =f(MX)∈Z[X]x∈Z
g(x)6∈ {0,1,−1}
k∈Zg(k) = 0 g(k) = 1 g(k) = −1
g g −1g+ 1
l l|g(x)M x f l
l|M
f(Mx) = Mnxn+· · · +Mx + 1 ≡1l
M
Φnp
Φn∈Fp[X]
n≥2{kn + 1 / k ∈N}
OΦn
ϕ(n)
mm m
p(Z/Zn)×
Φnm= 1
mΦnmathbbFpm= 1
p≡1nM
f∈Z[X]n
p f ∈Fp[X]Fp[X]

OK=Q[x1,· · · , xn]f A =
Z[x1,· · · , xn]K z ∈A K =Q[z]d≥1
dxi∈Z[z] 1 ≤i≤n
P=pz,Qz P ∈Z[X]
S p > d f ∈Fp[X]P∈Fp[X]
FpS
p∈S ζ ∈FpP
ϕ:Z[z]−→ Fpϕ(z) = ζ
d6= 0 Fpϕ
ϕ:Z[1
d][z]−→ Fp
Z[z]
j
ϕ//Fp
Z[1
d][z]
ϕ
==
1≤i≤n xi∈Z[1
d][z]⊂K f(xi) = 0
ϕ(f(xi)) = f(ϕ(xi)) = 0
ϕ(xi) 1 ≤i≤n f Fp
ϕ(xi) = ϕ(xj)ϕ(xi−xj) = 0 i6=j xi−xj∈(ϕ)
X(f)∈Z∩(ϕ) = Zp
p|X(f)f∈Fp[X]ϕ(xi) 1 ≤i≤n
fFpfM