141 - Polynômes irréductibles à une indéterminée

A K
P∈A[X]A[X]>1
A[X]
uP u ∈A∗
u u ∈A∗.
K[X]
>1K
(ii) (X2+ 1)2Q
2 3
C1
R1 2
Z[X] 2X
k K P ∈k[X]
P K[X]k[X]
P k[X]K[X]X2+ 1 ∈R[X]
C[X]
P∈K[X]L K
P K α ∈L L =K(α)P(α)=0
P∈K[X]L=K(α)P
K L =K[X]/(P) [L:K]
α K
C=R[X]/(X2+ 1) X2+ 1 Q(3
√2) = Q[X]/(X3−2)
X3−2
P P
Q(3
√2) j3 √2j23 √2X3−2

P∈K[X]P
K L K
L[X]P1i.e. P L
Li.e. P L
P∈K[X]P K
DK(P) [DK(P) : K]6n!
DQ(X3−2) = Q(3
√2, j) 6 = 3!
DQ(X4−1) = Q(i) 2 6= 4!
A
A[X] Frac(A)[X]
P=anXn+. . . a0∈A[X]P c(P)
c(P) = (a0, . . . , an)A∗P c(P)=1
P, Q ∈A[X]c(P Q) = c(P)c(Q)
P∈A[X]A[X]
p∈A A
>1 Frac(A)[X]
A A[X]
K= Frac(A)P=anXn+···+a0∈A[X]p∈A
p aianp2a0P
K[X]A[X]c(P)=1
p P =Xp−1+··· +X+ 1 Z
P(X+ 1) p
a=pα1
1. . . pαr
r∈Zr>2i αi= 1 Xn−a
Z
K= Frac(A)I A B =A/I B
L P =anXn+··· +a0∈A[X]an6= 0 B
P B L P K A[X]c(P)=1
X3+ 462X2+ 2433X−67691 Z2
X3+X+ 1 F2
p∈ZXp−X−1Z
Fp

X4+ 1 Z
Fpp
P∈K[X]n > 0P K P
L K [L:K]6n
2
X4+X+ 1 F2F2
F4=F2[j]j2+j+ 1 = 0
P∈K[X]n L m
(m, n) = 1 P L
X3+X+ 1 Q Q(i)
X3+X+ 1 Fpp= 2 5 Fp
Fpn3n
p r ∈N∗q=pr
q
Xq−XFpFq
rFpFq
Fp
P r FpP Xpr−XFp
Iq(n)nFq
Nq(n)n>1
Xqn−X=Y
d|nY P∈Iq(d)
P Nq(n) = 1
nX
d|n
µn
dqd,
µ
28 Nq(n)Nq(n)>0
Fpr=Fpr= 1
P r FpL=Fp[X]/(P)
q=pr

α∈L K
K
α K α
I{0}K[X]Mα∈K[X]
I= (Mα)Mαα
P∈K[X]α K P
α P P (α)=0 P K[X]
α K [K(α) : K] = deg(Mα)Mα
α K α [K(α) : K] = +∞
L K
Li.e. α ∈L L =K(α)
α∈Q(3
√2, j) = DQ(X3−2) Q(3 √2, j) =
Q(α)α=3
√2 + j
n∈N∗ΦnnCΦn=
Qω∈U∗
n
(X−ω)U∗
nn
Φn∈Z[X]
ΦnQ Z c(Φn) = 1 Φn
ω∈U∗
n[Q(ω) : Q] = deg(Φn) = ϕ(n)
ω∈U∗
nω0∈U∗ m(n, m)=1 Q(ω)∩Q(ω0) = Q
λn + 1 λ∈N∗
Q