Feuille d`exercices 4

n≥1nZ Z
A A
A
Z/2Z Z Z
M A M
{xi}i∈IM{rj}j∈J
A(I)→M ei7→ xiA=Z[X]
A I = (2, X)⊂A
A K A
K K A
V K (vi)i∈IV
A V A
A V K
K A A =K
K A A =K
V W K A V
W K
A
AZ ZN
M A N M N M/N
M
A=Z/6Z2A3A2A⊕3A
1
A M

A M n ∈NM
An
A
M A M
ZNZ(N)
A
Ann∈N
M A N M N M/N
M
Q Z
Q0 1 Q
Q/Z
(ei)i∈IQj∈I(ei)i∈I\{j}
A M A x ∈M
a∈A a 6= 0 ax = 0 M
M M Ann(M)
A=ZM
M
A A
A
Z/2Z Z Z
B A B
{xi}i∈IB{yj}j∈J
A[Xi]i∈I→B Xi7→ xiA k
k X k
1

A[X]
A A[X]A
A a, b, . . . A
pgcd(a, pgcd(b, c)) = pgcd(a, b, c)
pgcd(an, bn) = pgcd(a, b)nn≥1
pgcd(a, b)ppcm(a, b) = ab
pgcd(a, ppcm(b, c)) = ppcm(pgcd(a, b),pgcd(a, c))
ppcm(a, pgcd(b, c)) = pgcd(ppcm(a, b),ppcm(a, c))
A=Zpgcd(a, b) = Pk|a,k|bϕ(k)ϕ
A δ a =bq +r
δ(r)< δ(b) pgcd(a, b) = pgcd(b, r)
A=Zpgcd(na−1, nb−1) = npgcd(a,b)−1
A=k[X]kpgcd(Xa−1, Xb−1) = Xpgcd(a,b)−1
A=Z+iZ⊂C
AC Z 2
A
N:A→NN(z) = |z|21
A A
N A u ∈C
z∈A|u−z|<1
1 + i1 + 2i A 2 5
p
p A
p
p= 2 p≡1 (mod 4)
⇔ ⇒
⇒
p A x2+ 1 = 0 A/pA
Z/pZ−1p p ≡ −1
(mod 4)

k A k[X]X
k k[X]Xii∈N, i 6= 1
k[X]
m>2XmA
Xnn∈NXmA
(X2, X3)A
m(Xm, Xm+1)A
pgcd(a, pgcd(b, c)) = pgcd(a, b, c)a=X2b=X5
c=X6pgcd(ca, cb) = cpgcd(a, b)a=X2
b=X3c=X3
X6