Partie I. Arithmétique des entiers de Gauss. Partie II. Réseaux.

Σ = x2+y2,(x, y)∈Z2
Z[i]
Z[i] = x+yi, (x, y)∈Z2
Z[i]C Z
Z⊂Z[i],∀(z, z0)∈Z[i]2:z+z0∈Z[i]zz0∈Z[i]
Z[i]Z
u6= 0 v u v
w∈Z[i]v=uw
uZ[i]u u
Z[i]u={wu, w ∈Z[i]}
∀z∈Z[i],|z|2∈N;∀(z, z0)∈Z[i]2,|zz0|2=|z|2|z0|2
n
n∈Σ⇔ ∃u∈Z[i]n=|u|2
Σ Σ
u6= 0 zZ[i]u z |u|2|z|2
Z
u v ∈Z[i]
uv = 1
1−1i−i
u|u|2= 1
z
v z |v|= 1 |v|=|z|
1 + i5 Σ
x∈Ra∈Z|x−a| ≤ 1
2
u6= 0 zZ[i]q r Z[i]
z=qu +r|r|2<|u|2
z
u
Z[i]
RZ[i]
∀(z, z0)∈ R2,−z∈ R z+z0∈ R

Z[i]i
c s r Z[i]Z[i]
∀z∈Z[i], c(z) = z, s(z) = i z, r(z) = iz
Rr
∀z∈ R, r(z) = iz ∈ R
n≥2SnTn
Sn={z∈Z[i] Re(z)≡Im(z) mod n},Tn={z∈Z[i] Im(z)≡0 mod n}
s◦c=−c◦s=r
u∈Z[i]Z[i]u
Ru∈Z[i]R=Z[i]u
n≥2SnTn
nSn
R0
u∈ R uR
u0∈ R
∀v∈ R, v 6= 0 ⇒ |u0|2≤ |v|2
R=Z[i]u0|u0|= 1
p a ∈ZRa
Ra=xp +yip +z(1 + ia),(x, y, z)∈Z3
Ra
Ra
R0=TpR1=Sp
a b Z
Ra=Rb⇔a≡bmod p
a6≡ 0 mod pRa
a0∈Zaa0≡1 mod p
c(Ra) = R−as(Ra) = Ra0
a0≡ −amod pRa
p a
∀(z, z0)∈Z[i]2,Im(zz0)∈Z
∀(u, u0)∈ R2
a,Im(u u0)≡0 mod p
Ra6=Z[i]
T S

Rau u0Im(u u0) = p
a∈Za2+ 1 ≡0 mod p
Ra
p
Pcp−1p
∃a∈Za2+ 1 ≡0 mod p
P0
c
P
Pc⊂Σ
n
vp(n)
P0
cn∈Σ
pΣp=x2+y2x y Z
λ µ Zλx −µy = 1 a=λy +µx
x y λ µ a
(λ2+µ2)p= 1 + a2p∈ Pc
p∈ P0
cp
p∈Σ⇔p∈ Pc⇔p
5+5i−3+4i
n∈Σp∈ P0
cn p2n
Σvp(n)
Pc
p > 2I=J1, p −1K
x∈I I −x p
I x0xx0≡1 mod p x0
I ./
∀(x, y)∈I2, x ./ y ⇔(x4+ 1)y2≡(y4+ 1)x2mod p
./
x≡ −xmod p I
x∈I x =x0
x=p−x0x∈I
a∈I a2+ 1 ≡0 mod p
a
(x4+ 1)y2−(y4+ 1)x2
./ x ∈I
p∈ Pcp≡1 mod 4