Irréductibilité des polynômes cyclotomiques dans Q[X]

ΦpQ[X]
ω=e2iπ
ppΦp=Xp−1+. . . +X+ 1 p
ΦpQ[X]I
ωQ[X] ΦpQ[X]
(X+ 1)p−1+. . . + (X+ 1) + 1
Q[X]
ΦpQ[X]
Qhe2iπ
pi={Q(ω), Q ∈Q[X]}
Q
I={Q∈Q[X], Q(ω)=0} I Q[X]
Q∈Q[X]7→ Q(ω)∈C Q[X]I
QI=QQ[X]
Φp(ω) = 0 (ω−1)Φp(ω) = ωp−1 = 1 −1=0 QΦp
Φp∈ I Q6= 1 ΦpQ= Φp
I= ΦpQ[X]
U= (X+ 1)p−1+. . . + (X+ 1) + 1 = Φp(X+ 1)
U=1−(X+ 1)p
1−(X+ 1) =(X+ 1)p−1 X
=Xp−1+µp
p−1¶Xp−2+...+µp 2¶X+µp 1¶∈Z[X]
U
p
A=anXn+. . . a1X+a0∈Z[X]p
(i)p a0
(ii)p a0, a1, . . . , an−1
(iii)p2a0
AQ[X]
(i) (iii) (ii)¡p
k¢
p1≤k≤p−1 1 ≤k≤p−1k!¡p
k¢=p(p−1) . . . (p−k+1)
p k!¡p
k¢k < p p k!¡p k¢
U
ΦpΦp=BC B, C Q[X] deg B < deg Φpdeg C <
deg ΦpU= Φp(X+ 1) = B(X+ 1)C(X+ 1) deg B(X+ 1) = deg B < deg U
deg C(X+ 1) = deg C < deg U U
Φp=Xp−1+. . . +. . . +X+ 1 Q[X]

Φpω(1, ω, . . . , ωp−2)
Q
p−1ω Q ∈Q[X]R Q Φp
Q(ω) = R(ω) Φp(ω) = 0
Q[ω] = V ect(1, ω, . . . , ωp−2)
(1, ω, . . . , ωp−2)Q Q[ω]
dim Qhe2iπ
pi=p−1
Qhe2iπ
piQ[ω]
P7→ P(ω)Q[X]Q C
CxQ[ω]R∈Q[X]
p−1x=R(ω) Φp
R(U, V )∈Q[X]2UΦp+V R = 1 ω U(ω)×0 + V(ω)x= 1
V(ω)x= 1 V(ω)∈Q[ω]x
Qhe2iπ
pip−1Q