Contenu et automorphismes de k(X) 1 Factorialité des anneaux de

k(X)
A
A[X]A[X1, . . . , Xn]
k k(X)
A A[X]
P∈A[X]c(P)
A
A P, Q A
c(P Q) = c(P)c(Q)
c(P)=1 P
A{ap}p∈PA[X]
{ap}p∈P
A[X]K[X]
Z[X]
Q[X]Z[X]
k(X)
k(X)
A K P ∈A[X]
c(P) = 1 P A[X]K[X]
P K[X]P=QR A[X]
K[X]Q R K Q
1 = c(P) = c(Q)c(R) = ac(R)Q=a
A P A[X]
P A[X]P=QR Q
R K[X]λ, µ ∈K×Q=λQ0R=µR0
Q0, R0A[X] 1 P=λµQ0R0
1 = λµ P =Q0R0P A[X]
Q0R0A Q R K 

GL2(k)→Autk(k(x))
a b
c d 7→ X7→ aX +b
cX +d
PGL2(k)k
k(X)
(2,2) k(X)
PGL2(k) Autk(k(x))
ϕ:k(X)→k(X)F=ϕ(X) = U(X)/V (X)
ϕ k(F) [k(X) : k(F)] = 1
F=U(X)/V (X)n= max{deg(U),deg(V)}
[k(X) : k(F)] = n n = 1
[k(X) : k(F)] = n X
k(F)n
P(T)d´ef =V(T)F−U(T).
P k(F)[T]
[k(X) : k]
|{z }
∞
= [k(X) : k(F)]
| {z }
fini
[k(F) : k]
F k k[F]
P(T) = V(T)F−U(T)k[T][F]
•1F
•P1F
k(X)[F]k[X][F]
k[T][F] = k[F][T]P(T)k[F][T]
k(F)[T]