Sommes de deux carrés et entiers de Gauss.

Q⊂N
Q={a2+b2|a, b ∈N}
n∈Q n 6≡ 3 mod 4
Z[i]Q
p p 6≡ 3 mod 4 p= 2
p≡1 mod 4
Z[i]d(a+bi) = a2+b2
Z[i]
Z[i]p∈Z Z
5Z[i] 5 = (1 + 2i)(1 −2i)
p∈Z Z[i]p
Z
pZp
Z[i]p p =ππ π Z[i]
π p Z[i]ππ p
p
pZ[i]
Z[i]/(p)'Fp[X]/(X2+ 1) Fp=Z/pZ
pZ[i]X2+ 1
Fp[X]

p
pZ[i]
−1Fp
G n m G
gm= 1 g∈G
G
(Z/m1Z)× · · · × (Z/msZ)m1|m2|. . . |ms
m n m =n G
F∗
p
Xm−1
−1
p= 3,5,7
p−1Fp
F∗
p4−1
2F∗
p
F∗
pp−1
−1p p = 2 p≡1 mod 4
p
pZ[i]
−1Fp
p= 2 p≡1 mod 4
n∈Nn∈Q
Zn=
pα1
1. . . pαk
k3 mod 4
4410 = 2×9×5×49
924 = 4 ×3×7×11