TD2 : Anneaux, idéaux, anneaux principaux et factoriels

k k[X]
k[X, Y ]k[T2, T 3]Z[X]
Z[i√5]
k k k[X, Y ]
XnY n > 0
A
(I, J)A√I+J=√I+√J
a b u A (a) = (b)a=bu
u∈A×
Q×(Z/3Z(X))×
A
I A p√I=√I
A I A/I
A
(An)n∈N
A A =Sn∈NAnA
k
k k[X, Y ]/(X2−Y3)k[T2, T 3]
k k[X, Y ]/(X2−Y)k[T]

a b
k[Ta, T b]k[X, Y ]
k k[X, Y ]/(XY −1) k[T]
Z[i√7] Z[1+√5
2]Z[X]
A x ∈A
x
(x)A/(x)
(x)A/(x)
A I
A/I A
I
J⊇I A A/J
A/I J/I
I A/I
A/I A
I
R[X]/(X2+X+1),R[X]/(X3−6X2+11X−6),R[X]/(X4−1),R[X]/(X5).
R[X]/(X5(X4−1))
A A
A[X]
A
A a b
A(a) = (b)u∈A×a=bu
A=Q[X, Y, Z]/(X−XY Z)x y z
X Y Z A
A

(x)=(xy)
u∈A×xu =xy
A=C[X, Y, Z, T ]/(XY −ZT )
B=Z[√10]
B up1...pnu∈B×pi
i
p q
Z[√pq]
R
x R
R×∪{0}R R/(x)
x φ(x)
x /∈R×φ
R
α=1+i√19
2A=Z[α]
A×
A
a, b ∈Ar{0}q, r ∈A r = 0
|r|<|b|a=bq +r2a=bq +r
A/(2)
A
Z[1+i√11
2]
(x, y)∈Z2y2+ 11 = x3
(x, y)y2+ 28 = x3
(x, y)∈C[T]2y2+T=x3
S={a2+b2|(a, b)∈Z2}
p−1
Z/pZ
p
n S vp(n)

Z[i]
p S
Z[i]
p S
n S
p vp(n)
a2+ab +b2
T={a2+ab +
b2|(a, b)∈Z2}
a2−2b2
S={a2−2b2|(a, b)∈Z2}
A=Z[√2]
p
2p
p≡1 mod 8 −1
Z/pZ2p
p
P∈Z/pZ[X]X4+ 1
Z/pZ[X]/(P)Z/pZ
−1
k=Z/pZ[X]/(P)α−1k
x k x2= 2 α+α−1
−α−α−1
x k Z/pZ
xp=x
2
Z/pZp
A
n n ∈S−n∈S
p
A
p S
S
n∈S(a, b)∈Z2
n=a2−2b2