M1 polynômes irréductibles Année 2015/2016

1 2015/2016
Q[X]
P=X4−6X2+X−1
P= 2X5+ 3X4+ 8X3−2X2+ 5X−1
P=X4+X3−X2+ 7X−1
P=X5−6X3+ 2X2−4X+ 5
P=X4+X3−2X2+ 4X−1Z[X]
P(1) = 3
P P F3[X]P
Z[X]Q[X]
−2 (F5)×
X4+ 2 F25
X4+ 2 F5[X]
X4+ 2 Z[X]
a∈CaQP a Q
AZ[X]a P A Z[X]

n Cn={x∈C|xn= 1}C∗
n
CnC[X]
Φn
Φn=Y
α∈C∗
n
(X−α)
Φn
Xn−1 = Qd|nΦdC[X]
Φn∈Z[X]
ω∈C∗
nP ω QE
PCp n E
p
p n
a E apE S ∈Q[X]Xn−1 = P S
S(ap) = 0
P S Z[X]
TZ[X]S(Xp) = P T
Z[X]Fp[X]
(S1(X))p=P1T1
γ P1Fp[X]S1
Fp[X]
ΦnZ[X]
E⊂C∗
n
s n ωs∈E
E=C∗
n
P= Φn
ΦnQ Z
k n Dn=det((Xi,j )1≤i≤n,1≤j≤n)
k[Xi,j ,1≤i≤n, 1≤j≤n]
n

n≥2A=k[Xi,j ,1≤i≤n, 1≤j≤n, (i, j)6= (n, n)]
K A DnA[Xn,n]
DnK[Xn,n]
Dn−1DnA
p P =Xp−X−1Z[X]
P P Fp[X]L P a P
L P L {a+i, 0≤i≤p−1}
PFp[X]
PZ[X]Q[X]
k f g k[X, Y ]f g
k[X, Y ]
Cf={(x, y)∈k2|f(x, y) = 0}Cg={(x, y)∈k2|g(x, y) = 0}Cf
Cgk2Cf∩Cg
˜
f˜g f g k(X)[Y]
˜
f˜g k(X)[Y]
f g k[X, Y ]
˜
f˜g k(X)[Y]
A Cf∩Cg
A={t∈k| ∃(x, y)∈Cf∩Cgt=x}A
Cf∩Cg