109: Anneaux Z/nZ. Applications. 1 L`anneau Z/nZ

Z/nZ
Z/nZ
Z
nZn∈N Z/nZ:= {a+nZ|a∈Z}=
{a+nZ|a∈ {0, . . . , n −1}} n n
a n Cn(a)
C6(0) = C6(2)C6(3)
Z/nZ
n∈N Z/nZo(Cn(m)) = n
n∧mZ/nZ,+)
m n ∧m= 1
ϕ(n)k6n k ∧n= 1
(Z/nZ,+)
p0p ϕ(p) = p−1
d n Z/nZ
d < Cn(n
d)>
ϕ
n=P
d|n
ϕ(d)
ϕ(1) = 1 ϕ(2) = 2 −ϕ(1) = 1 ϕ(3) = 3 −ϕ(1) = 1 ϕ(4) = 4 −ϕ(2) −ϕ(1) = 2
ϕ(5) = 5 −ϕ(1) = 4 ϕ(6) = 6 −ϕ(1) −ϕ(2) −ϕ(3) = 3
Cn(k) (Z/nZ,∗)⇔k∧n= 1 ⇔k(Z/nZ,+)
ϕ(n)
n
47 111 26
x∧n= 1 xϕ(n)≡1[n]
Z/nZ⇔ ⇔ Z/nZ Z/pZ
p

x∧n= 1 Z/nZ Z/nZ
Cn(d)d n d
p1, . . . , pn
Z/p1p2. . . pn'Z/p1Z×Z/p2Z×. . . ×Z/pnZ Z/p1p2. . . pn'
Z/p1Z×Z/p2Z×. . . ×Z/pnZp1, . . . , pn
Z/6Z'Z/2Z×Z/3Z
p1, . . . , pn
(Z/p1p2. . . pn)∗'(Z/p1Z×Z/p2Z×. . . ×Z/pnZ)∗
x≡a[n]
x≡b[m]
n∧m x0
x0+nmt t ∈Z
u n m
a+n(b−a)u
x≡10[47]
x≡5[111]
4334 + 5217t
ϕ(n)
ϕ∀(m, n)∈N2ϕ(mn) = ϕ(n)ϕ(m)
n∈N∗n=Q
i
pαi
iϕ(n) =
Q(pαi
i−pαi−1 i)
ϕ(21060) = ϕ(4 ∗13 ∗5∗81) = (22−2)(13 −1)(5 −1)(34−33)=2∗12 ∗4∗54 = 5184.
Z/nZ
Aut(Z/nZ)'(Z/nZ)∗.
p α ∈N∗) (Z/pαZ)∗'Z/ϕ(pα)Z
(Z/2Z)∗'1α>2 (Z/2αZ)∗'Z/2Z×Z/2α−1Z
3Z/nZ
Aut(Z/nZ) (Z/nZ)∗n= 2 n= 4 n=pα
n= 2pαp α ∈N

n akak−1. . . a0
n a02
n a0+a1+. . . +ak3
n a05
n a0+ 3a1+ 2a2−a3−3a4−2a5+. . . 7
n a0+a1+. . . +ak9
n a0−a1+. . . + (−1)kak11
n a0−3a1−4a2−a3+ 3a4+ 2a5+a6+. . . 7
f:Z→A
n7→ n1A.
fZIm(f)'Z/nZn∈Nn
A n = 0 f A
Zk1A= 0 ⇔k= 0 n6= 0
AZ/nZk1A= 0 ⇔n|k
Z/nZA
Mn,m(Z/nZ) = n
A A 0
k0
kQ
p k Z/pZ
k k
k p ∃n∈N∗|k|=pn
k⊂L
L k
G2
>2a1|a2|. . . |anG'
n
Q
i=1
Z/aiZ
8Z/8Z Z/2Z×Z/4Z Z/2Z×Z/2Z×Z/2Z

P(x, y, z) = 0 P
x, y, z)
(x, y, z)∈N3x2+y2=
z2d∈Nu v
(x, y, z) (y, x, z) (d(u2−v2),2duv, d(u2+v2))
x= 0
y= 0
x4+y4=z2
x4+y4=z4
x3+y3=z3
xn+yn=zn
n>3
N
150 200
N E
p q N := pq
ϕ(n)=(p−1)(q−1) A eAϕ(N)
dAeAϕ(N)eAN ϕ(N)
dA
A
p q
N M A X
N MeBB
XdB
P∈Z[X]P=anXn+. . . +a0p
p6 |an
p|aii6n−1
p26 |a0
Q Z P
Xp−1+. . . +X+ 1 Z
p P ∈Z[X]P=anXn+
. . . +a0¯ P=Cp(an)Xn+. . . +Cp(a0)¯ PZ/pZ[X]
Q[X]Z[X]
X3+ 462X2+ 2433X−67691 Z Z/2Z[X]
p Xp−X−1Z Z/pZ

n>2∃a∈Za(n−1) '1[n]∀k < n −1,
ak6' 1[n]n
n>2n=pα1
1. . . pαk
k
∀i, ∃ai∈Za(n−1)/pi
i'1[n]∀k < n −1, ak6' 1[n]n
p h ∈ {1, . . . , p −1}n:= 1 + hp2
2n−1'1[n] 2h6' 1[n]n
p= 2127 −1h= 190 1 + 190p2
Z/nZ