TD13 : Extensions séparables

Fp(Xp, Y p)Fp(X, Y )
α∈Fp(X, Y )Fp(X, Y ) = Fp(Xp, Y p)(α)
K=F2(X, Y )L=K(α)α2+Xα +Y= 0 M=K(β)
β2=α KsK M
[M:K]=4
Ks=L[Ks:K]
γ∈M γ2∈K
γ∈K
K(F⊆M
K
L/K p > 0
p
F⊆E p > 0
x∈E
F(x) = F(xp)
(x1, . . . , xn)E F (xp
1, . . . , xp
n)
F E
(y1, . . . , yn)E F (yp
1, . . . , yp
n)
F E
(i)F⊆E

K=Q(3
√2, ρ)t∈Q3 √2 + tρ
K/Q3
√2 + tρ 3 √2
Q(√2,√3,√5)/Q
p
F⊆E p > 0
E×p⊆F×F⊆E
E(x1, ..., xn)E
E=F(x1, ..., xn)
E[E:F]
p E/F
K p > 0K K
KsK K
Ks
P∈K[X]
P K
r∈Na∈K P =Xpr−a
K⊆L
K⊆L
K L K
L Ks∩L
L L x ∈L
r∈Nxpr∈K
K K
L(K L
n q
Fq(X1, ..., Xn)
M p n
a∈M a p M

Xpn−a M
L/K L
[L:K]<∞K[L:K] = ∞
L K
p > 0 [L:Lp]<∞
L/K
K[X]L
L K
K[X]L
K
K
p > 0M={x∈L|∃n∈N∗, xpn∈K}
L
L
M[X]L
Fp(t1/p)⊗Fp(t)Fp(t1/p)