TD 2 : Anneaux d`entiers de corps de nombres

(a)α∈CK=Q(α)
αQ
P=Y
σ
(X−σ(α))
σ K Cα
Q(α)
(b)n∈N∗C > 0
αCn|σ(α)| ≤ C σ
Q(α)C
(c)α∈CNQ(α)/Q(α) = ±1α
(a)Q[√−3] Z[√−3]
(b)
(c)
(d)
A, B A ⊂B
b∈B A P ∈A[X]
(a)b∈B A
B A B
(b)B A
(c)K K OK
OKA⊂KZ
OK
K=Q(√7,√10) α∈ OKPQ
QZ[X]QF3[X]
(a)Q(α)Z[α]P Q F3[X]
Z[α] = OK

(b)α1= (1+√7)(1+√10) α2= (1+√7)(1−√10)
α3= (1 −√7)(1 + √10) α4= (1 −√7)(1 −√10) αiαj
i6=jZ[α]αn
i
(c)Pi∈Z[X] 1 ≤4Pi(α) = αiP
PiPji6=j Pi
ni
P Pjj6=i Pi
(d)P
OKZ[α]
OK
K α ∈ OKOK=Z[α]p
OK
(a)P α QP=Qr
i=1 Pi
e
i
p
Pi∈Z[X]PiOKp
(p, Pi(α))
(b)pOK=Qr
i=1(p, Pi(α))e
i
(c)d∈Z
p pOQ(√d)