Corrigé du TD13

Fp(Xp, Y p)Fp(X, Y )
α∈Fp(X, Y )Fp(X, Y ) = Fp(Xp, Y p)(α)
Fp(Xp, Y p)(X+XpkY)k≥1
pFp(Xp, Y p)X+XpkY /∈Fp(Xp, Y p)
(X+XpkY)p∈Fp(Xp, Y p)
i6=j X +XpiY X +XpjYFp(X, Y )Fp(Xp, Y p)
Y= (Xpi−Xpj)−1((X+XpiY)−(X+XpjY))
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
F⊆E p > 0
x∈E
F(x) = F(xp)
x F P =Xp−xp∈
F(xp)[T]
x F (x)P
x∈F(xp)F(x) = F(xp)
F(x) = F(xp)Q∈F[T]x=Q(xp)x
Q(T)−T x
F

(x1, . . . , xn)E F (xp
1, . . . , xp
n)
F E
(y1, . . . , yn)E F (yp
1, . . . , yp
n)
F E
(y1, ..., yn)F E
A, B ∈GLn(F)
A
x1
xn
=
xp
1
xp
n
, B 
x1
xn
=
y1
yn
.
F r E
yp
1
yp
n
=F r 
y1
yn
=F r(B)F r 
x1
xn
=F r(B)A
x1
xn
.
F r(B)A∈GLn(F) (yp
1, ..., yp
n)F E
(i)F⊆E
x∈E n = [F(x) : F]
E(xiωj)1≤i≤n,1≤j≤[E:F]
n
ω1= 1
(xpiωp
j)1≤i≤n,1≤j≤[E:F]
n
F E
(1, xp, x2p, ..., x(n−1)p)F F (x)
F(x) = F(xp)x
F
E/F x ∈E
E=F(x)F E
(1, x, x2, ..., x[E:F]−1)E=F(x) = F(xp)
(1, xp, x2p, ..., xp[E:F]−p)F E

K=Q(3
√2, ρ)t∈Q3 √2 + tρ
K/Q3
√2 + tρ 3 √2
t∈Q3
√2+tj
K/Qt(j−j2) = y−3
√2y∈ { 3 √2, j 3 √2, j23 √2}
tQ\ {0}
t∈Q3
√2 + tj 3 √2
K/Qt(j−x) = y−1x∈ {1, j2}
y∈ {j, j2}Q(√2,√3,√5)/Q
K=Q(√2,√3) L=Q(√2,√3,√5)/Q
√2 + √3K
√2 + √3 + √5L
p
F⊆E p > 0
E×p⊆F×F⊆E
E(x1, ..., xn)E
E=F(x1, ..., xn)
E[E:F]
p E/F
(x1, ..., xn) 0 ≤j≤n
Ej=F(x1, ..., xj)j≤n−1xj+1 6∈ Ej
(x1, ..., xn)xp
j+1 ∈F
Xp−xp
j+1 ∈Ej[Ej+1 :Ej] = p
[F:E] = pn
E
K p > 0K K
KsK K
Ks
Ks
L M K
x M K[X]
x L L ⊆LM
K⊆L[L:K] = |K(L, K)|
K⊆L⊆LM
P∈K[X]
P K
r∈Na∈K P =Xpr−a

P Xpr−a α P K
K P = (X−α)pr
P α K P
(X−α)kα∈K P k = 1 P
P∈K[Xp]k=pk1
k1≥1P= (Xp−αp)k1P1= (X−αp)k1
K k =pr
K⊆L
K⊆L
K L K
K⊆L
K L K x ∈L
x x
K P Xpr0−a0P
K P =Xpr−a
K K(x)K
K⊆L
K(L, K) = {1}L→K x ∈L
K K(x)→K
x K K
L Ks∩L
Ks∩L
LK(L, K) = K(Ks∩L, K)
(Ks∩L)L K
K(L, K)→K(Ks∩L, K)
x /∈Ks∩L L →K
x x
Ks∩L P := Qϕ(X−ϕ(x))
ϕ L →K x
x P
M(Ks∩L)(x)ϕ M
M=Ks∩L x
M)Ks∩L x
Ks∩L⊆L
L L x ∈L
r∈Nxpr∈K
K K
x7→ xpK
P=anXnp +an−1X(n−1)p+··· +a0
bp
i=aiP= (bnXn+bn−1Xn−1+··· +b0)p
K
K⊆K

L(K L
p > 2
K=Fp(t)⊆M=DK(X2−X+t)⊆L=DM(Xp−α)⊆K,
α∈M X2−X+t∆=1−4t /∈K×2∪{0}
M2K L =DM(Xp−α)p Xp−α
M L
p K ⊆L
K(L, K)Id σ :x7→ t1/p
xx X2p−Xp+t
L(L, K) = LL
L⊆Lσ=Kx+t1/p
xx+t1/p xx+t1/p xp
=
xp+t
xp= 1 x+t1/p x= 1 L K L
L
K⊆L
n q
Fq(X1, ..., Xn)
K L =Fq(X1, ..., Xn)p
n > 0q=pnK
p > 0Kp=K K Tr≥0Lpr⊆
Fq[X1, ..., Xn]×∪ {0}=Fq
Fq(X1, ..., Xn)Fq
K=F3(X, X1/3, X1/32, ...)
3K3=K
T2−X K
M p n
a∈M a p M
Xpn−a M
L/K L
[L:K]<∞K[L:K] = ∞