111: Anneaux principaux. Applications

A∗
(a)a1A+a2A+. . . anA
(a1, . . . , an)
X A
X X
nZmZnmZ
nvmZ
A/I
Z[X]
Z/4Z0
Z/4Z

Z[X]
ϕ:A\0→Na=bq +r
ϕ(r)< ϕ(b)
Z
K K[X]
Z[i]k[[X]]
Z[√d]d= 2,3,7,11
A[X]A
⇒A
A∗∪0A A/(x)
Z[1 + i√19
2]
Z[1 + i√19
2]
A[X]
a|b
a|b⇔(b)⊂(a)
∃q∈A∗a=qb

p p|ab ⇒p|a p|b
Z
p=ab ⇒a∈A∗b∈A∗
Z
K[X]
K∗
A/I
A/(ab)'A/(a)×A/(b)
Z[i√5] 9 = 3 ∗3 = (2 + i√5)(2 −i√5)
P
Z K[X]
Z[X]
p|ab p|a p|b
a|bc a b a|c

d
Z[i√5] 2 + i√5 3(2 + i√5)
dA =aA +bB m mA =aA ∪bA
K[X, Y ]
K[X, Y ]
uQ
P
pv(pi)
iv(pi)
piv(pi)
A∩B
Z
1−1 1 2
D d1|d2. . . |dn
did0
i
di

(1,1, . . . 1)
M
L
L C
M
A
XIn−A
pFp
Fp
x2−a= 0 Fp
x2−5y2x y
−1K