Corps finis, corps de fractions rationnelles (TD4)

Fq
p q p n ≥d≥1f∈Fq[X0, . . . , Xn]
d
Pf(x0, . . . , xn)q−1(x0, . . . , xn)Fn+1
q
f(0, . . . , 0)
Fq
p q p d ≥1
dFp
Fprr≤d
2
F45F2[X]
FqFqd
dFq[X]
Fq
p n ≥1
Fp⊆Fpn
Fp⊆Fpnn
Fr : x7→ xp
m≥1
FpnFpmm n
Fpnpm{x∈Fpn|xpm=x}
m n
Fpm⊆Fpnn
m
Frm
Gal(Fpn/Fp)
Fpn
Fq
p q p r ≥1
µpr(Fq)prFq
Fq
n≥1 Φn∈Z[X]nCΦ(p)
n
ΦnpFq
Φ(p)
nn
Fq
n p
(Z/nZ)×Φ(p)
nFp

Φ8p p
Φ(p)
nFqq(Z/nZ)×
K
L=F∈K(X)
F(X) = F(1/X).
k n ≥1K=k(X1, . . . , Xn)
SnK Xi
KSnK
k=CGGLn(C)
K
GZn
Zrr≤n KGK
KGK
K F ∈K(X)rK F =P
QP, Q ∈K[X]
F K
X K(F)P(T)−F Q(T)∈K(F)[T]
K(F)⊆K(X)δ(F) = max(deg P, deg Q)
K(X)K
PGL2(K)
K L K K(X)
[K(X) : L]
n= [K(X) : L]F=P
Q∈LrK P, Q ∈K[X]
δ(F) = max(deg P, deg Q)
δ(F)≥n L =K(F)
F∈LrK δ(F) = m F =P
Q
P m
M(T) = P(T)−F Q(T)∈L[T]
L=K(F)
M(T)L[T]
Ψ∈L[T]∩K[X, T ]
P(T)Q(X)−Q(T)P(X)L[T]∩K[X, T ]
XΨ∈K[T]
ΨK
L=K(F)