116: Polynômes irréductibles. Corps de rupture

A A[X]A[X]
P A[X]P
A1C
RX
Z
P B A
A P A B =F r(A)
2XZ
Q2X2−2Q
R(X2−2=(X−√2)(X+sqrt2) X2+ 1 R
C
K P ∈K[X]
K L K L =K(α)P(α)=0
P K
K[X]/(P)
X2+ 1 R
R[X]/(X2+ 1) = R[i] = C
X3−2Q
Q[X]/X3−2 = Q[3√2] Q

Q[X]/X3−2 = Q[3√2] X3−2
3√2j3√2j2
P K L K P
P
n!
X3−2QQ[3√2, j]
Q[3√2] Q[j] 3
X4−2QQ[4√2, i]
X4+ 1 R C
CX4+ 1
C
X4−1QQ[1, i]
A
C(P Q) = C(P)C(Q)
A[X]
A[X]
P∈Z[X]P=anXn+. . . +a0p
p6 |an
p|aii6n−1

p26 |a0
Q Z P
Xp−1+. . . +X+ 1 Z
P(X+ 1)
I
P∈A[X]P=anXn+. . . +a0¯ P=CI(an)Xn+. . . +CI(a0)¯ P
A/I F r(A/I)F r(A)[X]A[X]
p Xp−X−1Z Z/pZ
X3+ 462X2+ 2433X−67691 Z Z/2Z[X]
X3+ 4X2−5X+ 7 Z Z/2Z[X]
X3−6X2−4X−13 Z Z/3Z[X]
X3+ 30X2+ 6X+ 1 Z Z/5Z[X]
X2+X+ 1 F2
F2(X2+X+ 1)2= 1 + X2+X4
5X4+ 17X3−X2−6X+ 23 Z Z/2Z[X]
X4+ 5X3−3X2−X+ 7 Z Z/2Z[X]
F3
X2+1 X2−X−1X2+X−1
F3
X4+ 7X24X+ 1 Z Z/3Z[X]
Z/2Z[X]
X4+ 4X3+ 3X2+ 7X−4 2,3,5,7et11
P
Z
X4+ 1 FpFp

p∈k(X]
(degP )/2
X4+X+ 1 F2X4+ 8X2+ 17X−1Z
k K
P P k K
X3+X+ 1 Q Q[i]
X4+ 1 Q Q[i]
q=pnq
Xq−XFp
Fq
FpnZ/pZ
F4'Z/2Z[X]/(1+X+X2)F8'Z/2Z[X]/(1+X+X3)F9'Z/3Z[X]/(X2+1)
F16'Z/2Z[X]/(1 + X+X4)
Q
k⊂K
α∈K
k α α
α α
k[X]µ(α)
α
K[α] = K(α)
K[α]
K[α]
K[α]'K[X]/(µ(α))
K k k

L K
∃α|K[α] = L
X4−2Q[4sqrt(2), i] = Q[i+4p(2)]
n k
k µn(k)
k∗
ϕ(n)µn(k)∗
Φn(X) = Q
x∈µn(k)∗
(X−x)
ϕ(n)
Xn−1 = Q
d|n
ϕ(d)
Q
σ:Z→k
k σ
Z Q
Xn−1
Q[X]
P hin(X) [Q(α) :
Q] = ϕ(n)
α β
Q(α)∩Q(β) = Q
an + 1