anneaux factoriels Exercice 1 : Exercice 2 : Exercice 3 : Exercice 4

A π1π2A
π1π2
a, b A a b
m A
d∈A ab =md
d a b
k P k[X]P
P0k[X]P
k[X]
PQ[X]
C
P Q Z[X]Q Q P
C[X]Q P Z[X]
C Z[i] = {a+ib;a, b ∈Z}
N:Z[i]→Nz∈Z[i]N(z) = zz =|z|2
Z[i]

z∈Cw∈Z[i]|z−w|2≤1
2
Z[i]
Z Z[i]
Z[i]
2 = −i(1 + i)22
Z[i]
p∈Z
Z[i]/(p)Fp[X]/(X2+ 1)
pZ[i]p≡ −1mod 4
p∈Zp≡1mod 4
πZ[i]π p Z[i]
ππ p Z
p=N(π)
Z[i]
πZ[i]
Z∩πZ[i]Z Z ∩πZ[i] = pZ
pZ
p=N(π)p= 2 p≡1mod 4
n
N
C={n∈N;∃a∈N,∃b∈Nn=a2+b2}
C
p∈Np∈C p
Z[i]
C
Z
E(x, y, z)x2+y2=z2
E=A
A={(d(u2−v2),2duv, d(u2+v2)) |(u, v, d)∈Z3}∪{(2duv, d(u2−v2), d(u2+v2)) |(u, v, d)∈Z3}.

E=ZF F ={(x, y, z)∈Z3|pgcd(x, y) = 1 x2+y2=z2}.
(x, y, z)∈F ω =x+iy ∈Z[i]
ωω =z2
ω ω Z[i]
δ ω ω Z[i]
δ δ Z[i]
N(δ)ωZ[i]
(x, y, z)F
λ∈Z[i]×α, β ∈Z[i]ω=λα2ω=λ−1β2
F⊂A E ⊂A
A π A
f:A[X]→A/(π)[X]Pi=d
i=0 aiXi
Pi=d
i=0 aiXif
A/(π)[X]
A K P Q
A[X]Q Q P K[X]Q
P A[X]
ACa+ib√5a b
Zz a N(z) = zz
z A a N(z) = 1
A×A
pZp
A a b Zp=a2+ 5b2
A2,3,5,7,11,13

v= 1 + i√5N(v)v A
A
A
2A A
I= 2A+vA A I
2v
k A =k[X, Y, Z, T ]/(XY −ZT )
A
f:k[X, Y, Z, T ]→k[U, T, Y ]
f(P(X, Y, Z, T )) = P(UT, Y, UY, T )
f
f f :A→k[U, T, Y ]
f
Y A
A
CR2ω C F
R2RI F
C M F ω FC
CRF/I Fω
ωRF/M M F
R2R
A=R[X, Y ]
CR2I
A C
C A/I
C y2−x2−2x−1=0

C C1C2
i= 1,2A Ci
I= (Y2−X2−2X−1)
A/I
C y2+x2−1 = 0
I= (Y2+X2−1)
A/I
C y2−x2−1 = 0
I= (Y2−X2−1)
A/I
C1y−x= 0 C2y−x−1=0
C=C1∪C2I1, I2I A
C1, C2C
p1:A→A/I1A/I
A/I →A/I1×A/I2
C1, C2C
A/I A/I1×A/I2
C]−1; 1[
I C 0
C
C