Feuille d`exercices 3 Exercice 3. Soit K = Q( Exercice 4. Soit K = Q

G G
G
K G K×G
Q
d∈Z[Q(√d) : Q]Q
√dCHomQ−alg(Q(√d),Q(√d))
a, b ∈Z√a∈Q(√b)a/b
K/Q[K:Q] = 2 K=Q(√d)
d
k2
k k k(√x)
x∈k×\(k×)2k(x) = k(y)x/y
K=Q(√5,√7) ⊂C
[K:Q]=4 K/Q
HomQ−alg(K, K)Z/2Z×Z/2Z
K
x∈K x Q(x) = K
QK√5 + √7
k√5 + √7K k
K=Qp2 + √2⊂Q
p2 + √2Q
HomQ−alg(K, K)Z/4Z
K
k p a ∈k
P(X) = Xp−X−a∈k[X]P
p= 2 k
k(x)x2−x∈k x /∈k x y k(x) = k(y)
(x−y)2−(x−y)∈k b2−b b ∈k
p p
Φp(X) = Xp−1+···+X+1 Q[X]
Φp(X+ 1)
[Q[e2iπ/p] : Q] [Q[cos(2π/p)] : Q]

P k[X]d L
k
[L:k]≤d!
d= 3
M F L, K ⊂M M
k= [K:F]l= [L:F]
[KL :F]≤kl
k l [KL :F] = kl
[KL :F]< kl
x F
[F(x) : F]F(x2) = F(x)
2 3 3
[F(x) : F]F(x3) = F(x)
k L =k(T)T
P, Q ∈k[T]U=P(T)/Q(T)∈
k(T)K k(U)L
T K
L K
P Q
Autk−alg(k(T)) PGL2(k)
L/k k
L