109 - Anneaux Z/nZ. Applications. 1 Généralités 2 Indicatrice d`Euler

Z/nZ
m, n ∈N∗m∧n=pgcd(m, n)
G n G
Z/nZ
Z/nZ
d∈N∗d≤n
dZ/nZd|n
Z/nZn
a b
u v au+bv = 1
m∧n= 1
Z/mnZ'Z/mZ×Z/nZ
(a, b, c)∈Z3ax +by =c x y
Z
a∧b c
(x0+bk
a∧b, y0−ak a∧b)k∈Z(x0, y0)
s∈Z
s∧n= 1
s(Z/nZ,+)
s∈(Z/nZ)∗

Aut(Z/nZ)→(Z/nZ)∗
φ7→ φ(1)
ϕ n ∈N
Z/nZ
ϕ(n) = |(Z/nZ)∗|=|Aut(Z/nZ)|
m∧n= 1
ϕ(mn) = ϕ(m)ϕ(n)p α
ϕ(pα)=(p−1)pα−1
(Z/nZ)∗n
n=Qipαi
i(Z/nZ)∗'Qi(Z/pαi iZ)∗
p α (Z/pαZ)∗'
Z/pα−1(p−1)Z(Z/2Z)∗' {1}(Z/4Z)∗'Z/2Z α > 2
(Z/2αZ)∗'Z/2α−2Z×Z/2Z
(Z/nZ)∗n= 4 n=pαn= 2pαp
pq
Z/nZ
G/Z G
p2
Z/pZ N
p
n p n
p
n
p=
0a≡0 [p]
1a p
−1a p
Zx2−p·y+n= 0

Z/pZ(p+ 1)/2
∀a∈Z,a
p≡a(p−1)/2[p]
N N =Qipαi
ia
a
N=Y
ia
piαi
a≡b[N]a
N=b Na N= 0
a∧N > 1
∀a, b ∈Z,ab
N=a Nb N
−1
N= (−1)(p−1)/22 N= (−1)(p2−1)/8
N, M
N
MM N= (−1)N−1
2
M−1
2
p p =x2−6y2
x, y ∈Zp y p x p2
p6≡(xy−1)2[p]6
p= 1 (−1)(p2−1)/8(−1)(p−1)/2p 3=
1p
p V
Fpu∈Gl(V)
uS(V)
ε(u) = det(u)
p
n≥2
Qp∈P pvp(n)
p≡3 [4] vp(n)

p q n =pq
d ϕ(n)
a∈Z/nZf:Z/nZ→Z/nZ, a 7→ ad
f g :a7→ aee
dZ/ϕ(n)Z
g e
ϕ(n)p q n
p q
n(n−1)! ≡ −1 [n]
a∈Za∧n= 1 aϕ(n)≡1 [n]
n an−1≡n[n]
n∀a∈Z:a∧n= 1, an−1≡a[n]
n∀a∈Z, an≡a[n]n
p n p −1|n−1
H={a∈(Z/nZ)∗/a(n−1)/2≡a
n[n]}n
H= (Z/nZ)∗
n k
1−2−k
ρ
N f :Z/NZ→Z/NZ
f(x) = x2+ 1 n x0∈
Z/NZxn+1 =f(xn)
n4
√N
xn−x2n∧N
N