Quelques remarques sur les anneaux Z/nZ 1 Structure de Z/nZ

Z/nZ
EGL(E)
Z/nZ
Z/nZ
Z/pαZ
Z/nZ
k
Z/nZ
GLr(Z/nZ) SLr(Z/nZ)
SLr(Z)→SLr(Z/nZ)
Z/nZ
Z/nZ
ab a +· · · +a b
Z/nZ
Z/nZ
A x ∈
A\ {0}x mx:A→A
mx1y∈A xy = 1 x
mxA
y xy = 0 x

n=pα1
1. . . pαr
rn
Z/nZ
Z/nZ'Z/pα1
1Z× · · · × Z/pαr
rZ
Z/pαZ
Z/pαZ
A=Z/pαZ
A p x ∈A
0≤xi≤p−1x=x0+x1p+x2p2+· · · +xα−1pα−1
A x ∈A∗x06= 0 x6∈ (p)
π:Z/pαZ→Z/pZx
A π(x)Z/pZf:R→S
(p)
A=A∗t(p)
A
0⊂(pα−1)⊂ · · · ⊂ (p2)⊂(p)⊂A .
p x ∈A
k≤α−1x∈(pk)x=pku u
A
Z/nZ
k
k≥2n≥2Z/nZ→Z/nZ
k p
n1p−1k
ϕ:Z/nZ→Z/nZϕ(x) = xk
n=pα1
1. . . pαr
rn
Z/nZ'Z/pα1
1Z× · · · × Z/pαr
rZ
ϕ ϕ(x1, . . . , xr) = (xk
1, . . . , xk r)ϕ
i k
Z/pαi
iZn=pα
x∈Z/pαZx ϕ(x)
(p)x(p)ϕ(x) = xk(pk)
ϕ p k
k α ≥2
p∈A ϕ α = 1 ϕ
ϕ0 0 (Z/pZ)∗
(Z/pZ)∗'Z/(p−1)Zϕ

Z/(p−1)Z
k k p −1
k= 2 Z/nZ
n= 2
Z/pZp
Z/pαi
iZn=pα
p≥3p= 2
p α ≥1
(Z/pαZ)∗2Z/pαZ
pα−1p−1
2
i i ≤α pi
Z/pα−iZ→Z/pαZ(Z/pα−iZ)∗2pi(Z/pαZ)∗2
pi(Z/pαZ)∗2pα−i−1p−1
2
p(Z/pαZ)∗'Z/pα−1(p−1)Z
2Z/pα−1(p−1)Z2p−1
2 2 (Z/pαZ)∗2
pα−1p−1
2
piZ/pαZ
pα−i
x=x0+x1p+· · · +xα−i−1pα−i−1pix=pi(x0+x1p+
· · · +xα−i−1pα−i−1) (Z/pα−iZ)∗2pi(Z/pα−iZ)∗2pi(Z/pαZ)∗2
pi(x0+x1p+· · · +xα−1pα−1)2xjpjj≥α−i
pi
pi(Z/pαZ)∗2(Z/pα−iZ)∗2
p A =Z/pαZ
1 + p−1
2(p+p3+· · · +p2β−1)
α= 2β
1 + p−1
2(1 + p2+· · · +p2β)
α= 2β+ 1
x∈A x =piu
0≤i≤α−1u6∈ (p)u A x

i u
A
A∗2tp2A∗2tp4A∗2t · · · t p2β−2A∗2α= 2β
A∗2tp2A∗2tp4A∗2t · · · t p2βA∗2α= 2β+ 1
p2kA∗2
0∈A A
1 + p2β−1p−1
2+p2β−3p−1 2+· · · +pp−1 2
α= 2β α 
kZ/nZ
Z/nZ
r≥1
GLr(Z/nZ) SLr(Z/nZ)
f:A→B
Mr(A)→Mr(B)r
f M = (mi,j )
f(M) = (f(mi,j )) f
f: Mr(A)→Mr(B)
det(f(M)) = f(det(M)) f
GLr(A)→GLr(B) SLr(A)→SLr(B)
Mr(A×B)'Mr(A)×Mr(B) GLr(A×B)'GLr(A)×GLr(B)
SLr(A×B)'SLr(A)×SLr(B) GLr(Z/nZ) SLr(Z/nZ)
n=pα
GLr(Z/nZ) SLr(Z/nZ)
|GLr(Z/pαZ)|=p(α−1)r2(pr−1)(pr−p). . . (pr−pr−1)
α= 1
k=Z/pZGLr(Z/pZ)
pr−1
pr−p
|GLr(Z/pZ)|= (pr−1)(pr−p). . . (pr−pr−1)
Z/pαZ→Z/pZx7→ x
x x

ν: GLr(Z/pαZ)→GLr(Z/pZ)
M∈GLr(Z/pZ)N∈Mr(Z/pαZ)
Mdet(N) = det(M)
det(N)N∈GLr(Z/pαZ)N M
H= ker(ν) Id +N N
p(Z/pαZ)'Z/pα−1Zr2
|H|= (pα−1)r2|GLr(Z/pαZ)|=|GLr(Z/pZ)|.|H|
|SLr(Z/pαZ)|=p(α−1)(r2−1)(pr−1)(pr−p). . . (pr−pr−2)pr−1
det : GLr(Z/pαZ)→(Z/pαZ)∗
x∈(Z/pαZ)∗
(x, 1,...,1)
|SLr(Z/pαZ)|=|GLr(Z/pαZ)|
pα−1(p−1) .
GLr(Z/pZ)
SL2(Z/nZ)
SLr(Z)→SLr(Z/nZ)
Z/nZ
Z
Z/nZ
ZGLr(Z/nZ)
±1 GLr(Z)
SLr(Z)→SLr(Z/nZ)
rSL1(Z)'SL1(Z/nZ)'1
r= 1 r−1A∈Mr(Z)
det(A)≡1 (n)
U, V GLr(Z)UAV
a1, . . . , amb=a2. . . am
W=
b1
b−1 1
1
1
, X =
1−a2
0 1
1
1
, A0=
1 0
1−a2a1a2
1
1
.