MPSI Colle 13 (4 au 8 janvier 2016) : Groupes, anneaux et corps

E ℓ
E φ :E×E−→ E
φ E
x∗y φ (x, y)
RℓZ Q R C
K[X]C0(R,R)RN
ℓR∗
+R∗ −
ℓP(E)
nKn[X]
n ℓ Kn[X]
Kn[X]n
X2XK2[X]X3
E ℓ EE
R3ℓ
A=a b
c d 
22(R)
A=a b
c d A′=a′b′ c′d′
A+A′=a+a′b+b′
c+c′d+d′A×A′=aa′+bc′ab′+bd′ ca′+dc′cb′+dd′
ℓ2(R)
E ℓ ∗ ∗ ℓ
∀(x, y, z)∈E3,(x∗y)∗z=x∗(y∗z)
ℓ
ℓR∗
+x∗y=xy
223= 64 2(23)= 256
(x∗y)∗z̸=x∗(y∗z)ℓ∗
E ℓ ∗ ∗ ℓ
∀(x, y)∈E2, x ∗y=y∗x
Rℓ
E ℓ P(E)
Kn[X]ℓ
K[X]ℓ
2(R)ℓ
E ℓ
EE
R3ℓ
−→
v∧−→ u=−(−→ u∧−→ v)
2(R)ℓ
0 1
0 0 ×2 0 0 1 =0 1 0 0   2 0 0 1 ×0 1 0 0 =0 2 0 0 
(E, ∗)ℓ x E
∀y∈E, x ∗y=y∗x
(E, ∗)
ℓ∗
Z D Q R C
0
P(E)∅E
K[X] 0K[X] 0
C1(R,R)
2(R) 0 2(R)=0 0
0 0 
I2=1 0
0 1 
RN
R3
(E, ∗)ℓ
x E
y E x ∗y=y∗x=

x y x ∗y=y∗x=
x x−1
Zn−n
D Q R C
R
Z2
K[X]−P
XK[X]
RNu−u
EEf
2(R)A=a b
c d 
−A=−a−b
−c−dA
ad −bc ̸= 0
A−1=d
ad−bc −b ad−bc
−c
ad−bc
a
ad−bc =1
ad −bc d−b
−c a 
P(E)A
A=E A =∅
x y E x ∗y
(x∗y)−1=y−1∗x−1
x E x−1x−1−1=x
EE
f g E E
f◦g(f◦g)−1=g−1◦f−1
A B
(A×B)−1=B−1×A−1
(G, ∗)ℓ(G, ∗)
äℓ∗
äℓ∗G
äg G ∗
ℓ∗(G, ∗)
(G, ∗)
∗ℓ
(Z,+) (D,+) (Q,+) (R,+) (C,+)
(Q∗,×) (R∗,×)R∗
+,×(C∗,×)
(Z,−)
(N,+) Nℓ
0
N
(Z,×)
(K[X],+) (Kn[X],+) KN,+(2(R),+) C0(R,R),+
(U,×)n>2 (Un,×)
EE,◦E E
E E
E σE
(S+,◦)S+
(K[X],×) (Kn[X],×)KN,×
(2(R),×)
2(R) ( 2(R),×)
(P(E),∩)E=∅
(P(E),∪)

R3,∧∧R3,∧
ℓ
8!×37×12!×210
43 252 003 274 489 856 000
(G, ∗)G(H, ∗)
H G
(G, ∗) (H, ∗)
G
äH⊂G
ä∈H
ä∀(h, h′)∈H2, h ∗h′∈H H ∗
ä∀h∈H, h−1∈H H
∀(h, h′)∈H2, h ∗h′−1∈H
(Z,+) (D,+) (Q,+)
(R,+) (C,+)
n∈N(Kn[X],+) (K[X],+)
n>2 (Un,×) (U,×)
(C∗,×)
(2(R),×) ( 2(R),×)2(R)
2(R) 1
({I2,−I2},×)
(2(R),×)
(G, ∗) (H, ♭)
f:G−→ H
∀(g, h)∈G×H, f (g∗h) = f(g)♭f (h)
φ:R2−→ E E u
un+2 =aun+1 +bun
(x, y)E u0=x u1=y
φ
∀(( x
y),(x
′
y
′))∈R2×R2, φ (( x
y)+(x
′
y
′))=φ(( x
y))+φ(( x y))
φR2,+(E, +)
f: (G, ∗)−→ (H, ♭)
f(G) = H
fker f G H
fker 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
(A, +,×)A ℓ
(A, +) 0A
∃1A∈A, ∀a∈A, 1A×a=a×1A=a1A
∀a∈A, 0A×a=a×0A=a0A
∀(a, b, c)∈A3, a ×(b+c) = a×b+a×c×
+
(A, +,×)×

(Z,+,×) (D,+,×) (Q,+,×) (R,+,×) (C,+,×)
C1(R,R),+,×
C1R
RN,+,×
(K[X],+,×)
K
(2(R),+,×)
n
a b A
a×b=b×aa b
∀n∈N,(a+b)n=
n
k=0 n
kak×bn−k=
n
k=0 n
kbk×an−k
(a+b)2=a2+a×b+b×a+b2
an−bna b A
a×b=b×a
∀n∈N∗, an−bn= (a−b)×
n−1
k=0
ak×bn−1−k= (a−b)×
n−1
k=0
bk×an−1−k
1A−ana A
∀n∈N,1−an+1 = (1 −a)×
n
k=0
ak
1−X3= (1 −X)1 + X+X2
X2+X+ 1 ¯
(A, +,×)
a A
a̸= 0A∃b∈A, b ̸= 0A, a ×b= 0A
(Z,+,×) (D,+,×) (Q,+,×) (R,+,×) (C,+,×) (K[X],+,×)
RN,+,×1+(−1)n
RR,+,×QC0(R,R),+,×
[0;π]×sin
[0;π]×sin×[π;2π]×sin
(2(R),+,×)A=0 1
0 0 
(A, +,×)
(Z,+,×) (D,+,×) (Q,+,×) (R,+,×) (C,+,×) (K[X],+,×)
RN,+,× RR,+,× C0(R,R),+,×(2(R),+,×)
Z
3Z,+,×
Z
6Z,+,×¯
2ׯ 3 = ¯ 0
(A, +,×)
a A
a̸= 0A∃n∈N, an= 0A
(A, +,×)
3(R)A=
012
003
000
A̸= 0 3(R)A3= 0 3(R)
x y
(x+y) (x×y)

(A, +,×) (A′,+,×)
A′A
(A, +,×)
(A′,+,×)A
äA′⊂A
ä0A∈A′
ä∀(a, b)∈A′2, a +b∈A′
ä∀a∈A′,(−a)∈A′
ä1A∈A′
ä∀(a, b)∈A′2, a ×b∈A′
(A′,+) (A, +)
(Z,+,×) (D,+,×)
(Q,+,×) (R,+,×) (C,+,×)
RN,+,× CN,+,×
(R[X],+,×) (C[X],+,×)
(Z[ ],+,×) (C,+,×)
(Q[ ],+,×) (C,+,×)
Q+Q
C
(2(R),+,×)
(2(C),+,×)
(A, +,×)A∗
A(A∗,×)
A
Q∗=Q\{0}R∗=R\{0}C∗=C\{0}
Z∗={1,−1}Z∗( Z\{0}
K[X]∗=K∗K[X]∗( K[X]\{0}
C0(R,R)∗R
C0(R,R)∗(C0(R,R)\{0}
2(R)∗=2(R)2(R)∗(2(R)\{0}
Q[ ]∗=Q[ ]\{0}Q√2∗=Q[√2]\{0}
Z[ ]∗={±1; ± } Z[]( Z[ ]\{0}
(A, +,×)A∗=A\{0A}
R C Q
Z
Z[ ]
Q[ ] Q√2
K[X]K
C0(R,R)R
CN
2(R)
Z
3Z
Z
nZn
(A, +,×)A
(A′,+,×)A
Q Q √2R
C
Q[ ] C