Généralités sur les anneaux (TD1)

A I
A/I A I
J⊇I A A/J A/I
J/I
A/I
A I
C[X],R[X]/(X2+X+ 1),R[X]/(X3−6X2+ 11X−6),R[X]/(X4−1).
R R C
A B f :A→B
f B A
I J A
IJ ⊆I∩J
XRnC(X, R)X
R
C(X, R)
C(X, R)
∗
A
A A A
A A
A A
k A M2(k)
A
A

k k[[X]] = Pi≥0aiXi
ai∈k i ≥0
X k
k[[X]] X
k[[X]]
A
A A[X]
Z[X]t∈Z[X]
t=uQipiu∈Z[X]×pi
Z[i] Σ = {a2+b2|a, b ∈N}
N:Z[i]→Z
a+ib 7→ (a+ib)(a−ib),
Z[i]
Σ
Z[i]
p
pΣpZ[i]
Z[i]/pZ[i]'Fp[X]/(X2+ 1) p
Σp= 2 p≡1 mod 4
n∈N r {0,1}Qppvp(n)
nΣvp(n)p≡3
mod 4
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
2A
A
k P, Q ∈k[X]p q P
QR(P, Q)kq−1[X]×kp−1[X]→kp+q−1[X]
(A, B)7→ AP +BQ
R(P, Q) = 0
Q