Exercices : Groupes, Anneaux et Corps - MPSI

(R∗
+,×)R∗ +R∗ −
(F(R,R),+)
◦
K=R C Kn[X]K
n n (Kn[X],+)
Kn
D D ={n
10k|n∈Z, k ∈N}(D,+)
(R,+)
(U,×) (C∗,×)
n(Un,×) (U,×)
EP(E)∪ ∩
CCC
C
◦CC
(CC,◦)a∈C∗b∈Cfa,b :C−→ Cfa,b (z) = az +b
fa′,b′◦fa,b
({fa,b;a∈C∗, b ∈C},◦)
f1, f2, f3f4R∗R∗
f1(x) = x f2(x) = 1
xf3(x) = −x f4(x) = −1 x
G={f1, f2, f3, f4} ◦
G=R∗×R⋆ G (x, y)⋆(x′, y′) = (xx′, xy′+y)
(G, ⋆)
(G, ⋆)
ABCD A B C D
ABCD
ABCD ABCD
ABCD D4
(D4,◦)D4
G Z (G)G
G Z (G) = {a∈G| ∀ g∈G, ag =ga}Z(G)
G Z (G)G

H K (G, ⋆)H∪K
H⊂K K ⊂H
K=R C K [X]K
(K[X],+,×)K[X]Kn[X]
D D ={n
10k|n∈Z, k ∈N}D
(R,+,×)
(F(R,R),+,×)
Z[i] = {a+ib |(a, b)∈Z2}(Z[i],+,×)
Q[i] = {a+ib |(a, b)∈Q2}(Q[i],+,×)
Q(√2)a+b√2a∈Qb∈Q
Q(√2)(R,+,×)
n p n p (Un,×)
(Up,×)
(G, ∗G) (H, ∗H)
f:G−→ H
∀(g, g′)∈G2, f (g∗Gg′) = f(g)∗Hf(g′)
f
f(G) = H
fker f G Hf∗
ker f={g∈G / f (g) = H}
ker f G
fker f={G}
f f G f
f=f(G) = {h∈H / ∃g∈G, f (g) = h}
f H
f f =H
Z[i] = {a+ib |(a, b)∈Z2}Q[i] = {a+ib |(a, b)∈Q2}
N:Z[i]−→ N∀z∈Z[i], N(z) = z¯z
(z, z′)∈Z[i]2, N(zz′) = N(z)N(z′)
zZ[i]⇐⇒ N(z) = 1
Z[i]
F(Q,+,×)F=Q
∗