Théorème de l`élement primitif - Agreg

L/K
L=K(a1, . . . , an)aiK
K a ∈L
L=K(a)
p
F∗
q
K
n∈N∗S(n)L=K(a1, . . . , an)K
(n−1) aiL/K
•S(1) S(2) L=K(a, b)b P
a K Q b K n =deg(P)m=deg(Q)
M P Q a1=a, . . . , anP M
b1=b, . . . , bmQ M b bi
γ a +cb
c∈K c ∈K a +cb 6=ai+cbji∈[1, n]j∈[2, m]
c K c =a−ai
bj−bK
c
γ=a+cb γ L K(γ) = K(a, b)
K(γ)⊂K(a, b)γ=a+cb b ∈K(γ)b Q ∈K(γ)[X]
R(X) = P(γ−cX)∈K(γ)[X]S Q R K(γ)[X]S
1b∈K(γ)S > 1S Q(X) = Qm
j=1(X−bj)
bjS j > 2bjR i ∈[1, n]γ−cbj=ai
c b ∈K(γ)
a=γ−cb ∈K(γ)K(γ) = K(a, b)S(2)
•S(n)n≥2S(n+ 1) L=K(a1, . . . , an, an+1)
K a2, . . . , an+1 L1=
K(a1, . . . , an)S(n)a∈L L1=K(a)L=L1(an+1) =
K(a, an+1)an+1 S(2) γ∈L L =K(γ)
p q Q(√p, √q) = Q(√p+√q)
a=√p+√qQ(a)⊂Q(√p, √q) (a−√p)2=q=a2−2a√p+p2
√p∈Q(a)√q∈Q(a)Q(a) = Q(√p, √q)

P Q
Gal(L/K)
Q(√p1,...,√pn) = Q(√p1+··· +√pn)pi