Sous-anneaux

d∈N
Zh√di=na+b√d|(a, b)∈Z2o
Zh√di(R,+,×)
Z[i] = a+ ib
(a, b)∈Z2
Z[i]
z∈Z[i] N(z) = |z|2
∀(z, z0)∈Z[i]2, N(zz0) = N(z)N(z0)N(z)∈N
Z[i]
A=nm
n|m∈Zn∈N∗,o
A(Q,+,×)
A=nm
2n|m∈Zn∈No
A(Q,+,×)
A
∀x∈A, ∃y∈A, xyx =x
A
Z={x∈A| ∀a∈A, ax =xa}
Z A
Z

Zh√di⊂R1∈Zh√di
x, y ∈Zh√dix=a+b√d y =a0+b0√d
a, b, a0, b0∈Z
x−y= (a−a0)+(b−b0)√d a −a0, b −b0∈Zx−y∈Zh√di
xy = (aa0+bb0d)+(ab0+a0b)√d aa0+bb0d, ab0+a0b∈Zxy ∈Zh√di
Zh√di(R,+,×)
Z[i] (C,+,×)Z[i] ⊂C1∈Z[i]
∀x, y ∈Z[i] x=a+ ib y =a0+ ib0a, b, a0, b0∈Z
x−y= (a−a0) + i(b−b0)a−a0, b −b0∈Zx−y∈Z[i]
xy = (aa0−bb0) + i(ab0+a0b)aa0−bb0, ab0+a0b∈Zxy ∈Z[i]
Z[i] (C,+,×)
N(zz0) = |zz0|2=|z|2|z0|2=N(z)N(z0)N(z) = a2+b2∈N
z=a+ ib a, b ∈Z
z z0N(zz0) = N(z)N(z0)=1
N(z), N(z0)∈NN(z) = N(z0)=1
z= 1,−1,i−i
A⊂Q1∈A∀x, y ∈A, x −y∈A xy ∈A
A(Q,+,×)
x∈A y ∈A xy = 1
x=m
n, y =m0 n0n, n0xy = 1 =⇒mm0=nn0m
U(A) = nm
n|m∈Z, n ∈N∗o
A⊂Q1∈A∀x, y ∈A, x −y∈A xy ∈A
A(Q,+,×)
x∈A y ∈A xy = 1
x=m
2n, y =m0
2n0m, m0∈Zn, n0∈N
xy = 1 =⇒mm0= 2n+n0
m
U(A) = ±2k|k∈Z
Z⊂A1A∈Z
x, y ∈Z a ∈A
a(x−y) = ax −ay =xa −ya = (x−y)a
a(xy) = xay =xya
x−y∈A xy ∈A
Z A
x∈Z y ∈A xyx =x
y∈Z z =xy2
xzx =x3y2=xyxyx =xyx =x
z∈Z a ∈A x3y2ay2
x3y2ay2=y2ay2x3
xay2=y2ax
az =za Z