116 - Polynômes irréductibles à une indéterminée. Corps - IMJ-PRG

R C
P∈C[X]
C C
C[X]
R[X]
A K
P=Pn
i=1 aiXi∈A[X]\ {0}
P c(P) = pgcd(a1,...an)
∀P, Q ∈A[X]\ {0}, c(P Q) = c(P)c(Q)
P∈A[X]A[X]
p∈A A
P≥1K[X]
A=ZK=Q
A[X]
A[X]

P∈A[X]≤3P
A
P=Pn
i=0 aiXi∈A[X]p∈A
p an∀i∈ {0, . . . , n −1}
p|aip2a0P K[X]
Xn+p Xn+p p ∈Z
Z[X]∀n≥1Xp−1+· · · +X+ 1 Z[X]p
P=Pn
i=0 aiXi∈A[X]I A
an/∈I L A/I π A A/I
π(P)L[X]P K[X]
Xp−X−1Q
K
P K[X]
L K P
∃α∈L P (α)=0 L=K[α]
K[X]/(P)P
L K α ∈L P
[L:K] = d P
L=K[α]
L P
i:K→K0
K[X]K0[X]P∈K[X]
P0=i(P)L L0P
K P 0K0x x0
ϕ L L0i ϕ(x) = x0
C R X2+ 1

P∈K[X]n > 0P
K L K [L:K]≤n/2P
L
X4+ 4X3−2X2−X+ 3 Z
P∈K[X]n L
m K m ∧n= 1 P L
X3+X+ 1 Q(i)Q
P∈K[X]n≥1
L K
P P L L K P
P
P
L P [L:K]≤n!
K p n p
p > 0KnXn−1K µn,K
Xn−1Kn
Knµ∗
n,K
µn,K n
n
Φn,K =Y
ζ∈µ∗
n,K
(X−ζ)
|µ∗
n,K |=ϕ(n)
Xn−1 = Qd|nΦd,K
Φn,K ∈K0[X]K0K
Φn,Q∈Z[X]
Φn,QZ[X]

FpZ/pZ
P n n
K p n > 0|K|=pn
p n
pnXpn−X
Fp
qFq
q=pnPd,q ={P∈Fq[X]/P
d d}
Xqn−X=Y
d|n
Y
P∈Pd,q
P
Xqn−X
Xqd−X
Fq
Fq
ΦnFq
P∈k[X]P(u)P πu