Capes : arithmétique et théorie des groupes,I

Z,+Z/nZ,+ (n∈N∗)
n∈N, n ≥2a∈Z/nZ Z/nZ,+
ordZ/nZ(a) = ppcm(a, n)/a =n/pgcd(a, n).
Z/nZ
Aut(Z,+) Aut(Z/nZ,+)
Z/nZ
U(Z/nZ) = {x∈Z/nZ| ∃y∈Z/nZ:xy = 1}
U(Z/6Z)U(Z/8Z)U(Z/nZ)Z/nZ
U(Z/nZ) = {x∈Z/nZ|(x, n) =
1}
ϕ:N∗→N:n7−→ |U(Z/nZ)|ϕ
(n, m) = 1 ⇒ϕ(nm) = ϕ(n)ϕ(m)
ϕ(pr)p r ∈N∗
n=pr1
1· · · prs
sϕ(n) = n(1 −p−1
1)(1 −p−1 2)· · · (1 −p−1
s)
d, n ∈Nd n g G
n
{gkn/d |k≤det (k, d) = 1}
d G
d G ϕ(d)
Pd/n ϕ(d) = n
X
d/n
(−1)n/d.ϕ(d) = ½0 si npair
−nsi nimpair
(x, n) = 1 n xϕ(n)−1
5124 72
n∈N∗Dna b
a2=bn= (ab)2= 1 n≥3
Dnaibj0≤i < 2
0≤j < n
Dnn−gone
Dn
x=µ0 1
1 0 ¶y=µe2iπ/n 0
0e−2iπ/2n¶
R
R
E D E R
E D

R E
D
R p, q n =pq, m =
ϕ(n) = (p−1)(q−1) R e < m m e
Zm:= Z/mZd ed ≡1(modm)n, e
p, q, m
n
f:Zn−→ Zn:x7→ xe
g:Zn−→ Zn:y7→ yd
g(f(x)) = x(modn)f(g(x)) = x(modn)
p= 3, q = 11 16,21,02,12,09,03
28,26,27,24,26,14
n= 3 ×11 = 33 m= 2 ×10 = 20 e= 13
(n= 33, e = 13) Z20 13−1= 17(mod20) 17
f(x) = x13(mod33) g(y) := y17(mod33)
16,21,02,12,09,03 04,21,08,12,09,27
28,26,27,24,26,14 19,05,03,18,05,20