Irréductibilité des polynômes cyclotomiques sur Z

n∈∗Φn[X]
n∈∗Φn∈[X]
n
Φ1(X) = X−1∈[X]
n∈∗k∈[|1, n|],Φk∈[X]
F= Φd
d|n,d6=n
F∈[X]F
FΦn=Xn−1
Xn−1 = F(X)P(X) + R(X)P, R ∈[X], R < F
[X]P= ΦnR= 0
P∈[X] Φn∈[X]
ζ n p n ω =ζp
n
f g ζ ω [X]f, g ∈[X]
[X] Φn=fα1
1. . . fαr
rfi[X]
Φn
Φn(ζ)=0 ζ fiQ
f=fi∈[X]
ω
f, g ∈[X], f, g Φn
f g
Φn=Q
ζ∈U∗
n
(X−ζ)U∗
nn
Xn−1 = Q
d|n
Φd
n d d|n
[X] : F(Φn−P) = R R F Φn−P= 0

f6=g
fg|Φn[X]g(ω) = g(ζp) = 0 ζ
g(Xp)∃Q∈[X], g(Xp) = f(X)Q(X)
f G(Xp) = f(X)S(X)+R(X)
S=Q∈[X]
f(X)|g(Xp) [X]
p¯
f(X)|¯g(Xp) = ¯g(x)ϕ¯ f
pϕ|¯g fg|Φn[X]¯
f¯g|Φn p[X]
ϕ2|Φn|Xn−1∃B∈p[X], Xn−1 = ϕ2B
nXn−1=ϕ(ϕ0B+ϕB0)ϕ|¯n6= 0 p6=n ϕ = 0
ϕ f =g
∀s∈∗,∀α f, ∀p1,...ps(p1. . . pn)∧n= 1 f(αp1...pn)=0
s∈∗
s= 1
s∈∗α f k =p1. . . psn
f(αp1...pn−1)=0
ps∧n= 1
f(αp1...pn)=0
n f f > ϕ(n) = Φnf|P hin
fΦn
f= Φn
P[X] [X]
P=p∈p
P[X]P