Anneaux principaux et factoriels

A p
Op p =ab p a a =pa0
p=pa0b a0b= 1 bM
A
p p A a
p p ab A p a p b
I= (p)
O3⇒2
2⇒1 (p)6=A p p =ab
p a a =pa0p=pa0b a0b= 1 b
1⇒3p I 6=A c J = (c)⊃I= (p)
I p J =A p
J=I I M
A
Op p ab a b
p
a b M
A
A
O1⇒2A A S
A
I=Aa a
a=xy x y a I ⊂
6=Ax I ⊂ 6=Ay
x y a =xy S =∅
p A p|xy p 6 |x p x
pu +xv = 1 y=puy +vxy p|xy p|y
2⇒1P A x ∈P x 6= 0 P6=A x
x=p1· · · prP
pi∈P i 1≤i≤r A pi
P P =A piP
FAF
I I

M I M M =A p p p 6∈ I
J={x∈A/px ∈I}A I J 6=I J
c I pc I =J
mp:x−→ px I I I 6={0} {0}
x∈I\ {0}x=pky k ≥1I⊂M= (p)p y
A y ∈I6={0}mpy p
F=∅AM
A
A K
OK A QK=Q[x]
x∈A pQ,x ∈Z[X]
Q Z xj1≤j≤m
x1=x σj:K−→ C Q σj(x) = xjz∈K
z=
m
P
i=1
ζixi−1ζi∈Q1≤i≤m1≤j≤m σj(z) =
m
P
i=1
ζixi−1
j
ζi1≤i≤m
(xi−1
j)1≤i,j≤mV=Q
i<j
(xi−xj)6= 0
∆ = (pQ,x) = V2∈Z
ζi=
m
P
j=1
Si,j σj(z)
VSi,j 1≤i, j ≤m
(xi−1
j)1≤i,j≤m
1≤i≤m∆ζi=V
m
P
j=1
Si,j σj(z)
z∈A1≤i≤m ni= ∆ ζi∈Q
Zni∈Zz=
m
P
i=1
ni∆−1xi−1
A
m
L
i=1
Z∆−1xi−1M
A
OP A x ∈P\ {0}pQ,x =xr+cr−1Xr−1+· · · +
c1x+c0∈Z[X]c06= 0 pQ,x(x) = 0 c0∈P P ∩Z
ZP∩Z=Zp p A/P Fp
A
PM