Théorème de transfert de Gauss (Perrin)

P=PaiXiA
P c(P)
c(P)=1 a∈A
aaic(aP ) = pgcd(aai) = apgcd(ai) = ac(P)
c(P Q) = c(P)c(Q)
P Q p =c(P)
q=c(Q)P0=P/p Q0=Q/q A
c(P0Q0) = 1 C(P Q) = C(pqP 0Q0) = pq =
C(P)C(Q)
P Q c(P Q)6= 1 A
p c(P Q)c(P) = c(Q)=1
i0j0i<i0j < j0p|aip|bjp6 |ai0
p6 |bj0c i0+j0P Q p|Pi+j=i0+j0aibi=
ai0bj0+Pi+j=i0+j0,i<i0j<j0aibip|ai0bi0p
A p|ai0p|bj0
A
K A K[X]
A[X]K[X]
A[X]
K[x]
p
P(X)Q(X)P Q
0p=ab a b A A∗[X]
A∗[X]A
Q(X)R(X)A[X]
Q R K[X]Q=a P
1 = aC(Q)a
A[X]
0
A P
1P P
c(P)P0c(P)P
K[X]P=QR Q R K
Q(X) = Pai/biXiR(X) = Pa0
i/b0 ia aib bi
a0b0Q=a/bQ0Q0R=a0/b0R0
a b a0b0
A[X]bdP =acQ0R0bd =ac
A bd/ac ∈A∗P=λQ0R0λ∈A∗P A
Q0R00A∗
A[X]

P K[X]P=fα1
1fαr
r
fαi
i=ai/bif0
if0 iA
c(QbiP) = c(QaiQf0αi
i)P P =uQf0αi i
P dP 0P0
A|X]d A
P(P)
p A[X]/(p)'
A/(p)[X]p
P
P K[X] = ∩A[X] = P A[X]
A[X]/P A[X]K[X]/P K[X]