TD n°1 : Anneaux, divisibilité et irréductibilité

Z[√−5] 2
A a ∈A1 + a A
A A[X]
A
Z/4Z[X]
K E
K[X]
K E
K E
Z
K K v :K∗−→ Z
x, y ∈K∗
v(xy) = v(x) + v(y)v(x+y)≥min(v(x), v(y)).
K=Qp vpQvp(p) = 1
K d
K x, y, z ∈K
d(x, z)≤max(d(x, y), d(y, z)).
v2(a)
P+∞
n=0 2n
K v A := {x∈K|v(x)≥0}
A∗=v−1(0)
1.
A A π A
A a =πnu
u∈A∗
A(πn), n ∈N
P(X) =
n
X
k=0
akXkZ
p an
p P (X)Z/pZ[X]
P(X)Z[X]
p anp
P(X)Z[X]

H(C)
H(C)
f∈ H(C)
AZ→AZ[X]→AQ→Q
R→R
n≥1Zn→Z Q →Z
G A
G→G
G=Z/nZn≥1