liste de théorèmes

n|ab n ∧a= 1 n|b
p q
∃u, v, pu +qv = 1
p
0 (Z/pZ∗,×)
x, y ∈Z/pZ, xy = 0 ⇒x= 0 y= 0
p q
Z/pqZ'Z/pZ×Z/qZ.
p q
(x[p], x[q]) x[pq] (x1, x2)7→ x1qv +x2pu
x6= 0 ⇔xp−1
2= 1[p]
p a a
p
•p a
•a p
k k2=a(p)
• −1a p
ab
p=a pb p
a=b(p)a
p=b p
1
p= 1

−1
p= (−1)(p−1
2)=1p≡1[4]
−1p≡3[4]
2
p= (−1)„p2
−1
8«= 1 p= 1 7( 8) −1p= 3 5( 8)
a
2= 1 a0
q
p=p q(−1)(p−1
2)(q−1 2)
ϕ
n n
n
ϕ(8) = Card{1,3,5,7}= 4
n a
n
aϕ(n)≡1[n]
ϕ
ϕ(2k) = 2k−1
ϕ(pk) = pk−pk−1= (p−1)pk−1
ϕ(Qpαi
i) = Qϕ(pαi i)
a
p≡ap−1
2[p]