Exercices - Page Personnelle de Jérôme Von Buhren

E
(P(E),∆,∩)
P(E)
E I P(E)
X∈IP(X)⊂I
(X, Y )∈I2X∪Y∈I
P(E)P(F)F∈P(E)
E
I={X∈P(E)|X}.
IP(E)
IP(F)F∈P(E)
(A, +,×)a∈A
fa:A→A, x 7→ ax.
f
A a ∈A a
0
B A B×=A×∩B
(A, +,×) (a, b)∈A2
1−ab ∈A×⇔1−ba ∈A×.
(A, +,×)
N={a∈A| ∃n∈N, an= 0}.
N A
(A, +,×)I, J ⊂A
R(I) = {x∈A| ∃n∈N, xn∈I}.
R(I)A I
R(I∩J) = R(I)∩R(J)R(I) + R(J)⊂R(I+J).
A=ZI=nZn∈NR(I)
(A, +,×)
N={a∈A| ∃n∈N, an= 0}.
N A
(A, +,×) (a, b)∈A2
ab ba
a1−a
a b a +b
(A, +,×)x2=x x ∈A
A

(A, +,×)x3=x x ∈A
A
e∈A e2=e a ∈A b =ea(1 −e)
b2ea =ae
x2A x ∈A
A
Z[i] = {a+ib ∈C|(a, b)∈Z2}.
Z[i] (C,+,×)
Z[i]
Z[i]
d∈N√d /∈N
Z[√d] = {a+b√d∈R|(a, b)∈Z2}.
Z[√d] (R,+,×)
a+b√d∈Z[√d]a2−db2=±1
Z[√d]
ZnZ
D
D(Q,+,×)
D
A
A=nm
n∈Q
m∈Z, n ∈2N+ 1o.
A(Q,+,×)
A
A
PA
(Q,+,×)
P(A) = p∈P
1
p∈A
A B (Q,+,×)
P(A) = P(B)⇒A=B.
PPA(Q,+,×)
P(A) = P
(A, +,×)A
1A(A, +) A n
nx = 0 x∈A
A n
A x 7→ xn

d∈N
Ad={(x, y)∈Z2|d|(y−x)}.
Ad(Z2,+,×)
A(Z2,+,×)
H={x∈Z|(x, 0) ∈A}(Z,+)
d∈NA=Ad
IZ2
I1={x∈Z|(x, 0) ∈I}I2={x∈Z|(0, x)∈I}.
I1I2Z
I=I1×I2
(Z2,+,×)
Z/nZ
¯
7x=¯ 2Z/37Z
x2+x+¯ 7 = ¯ 0Z/13Z
x2−¯ 4x+¯ 3 = ¯ 0Z/12Z
(i)x≡1 [6]
x≡2 [7] (ii)3x≡2 [5] 5x≡1 [6] (iii)x≡2 [10] x≡5 [13]
x+y≡4 [11]
xy ≡10 [11]
(Z/20Z)×Z/2Z×Z/4Z
n∈Nn>3
a∈Na2n−2≡1[2n]
(Z/2nZ)∗
n∈Nn>2
ϕ(n)>nln 2
ln n+ ln 2.
ϕ(n)n∈Nn>3
n∈N∗
d∈Nd|n
{k∈J1, nK|k∧n=d}.
n=X
d|n
ϕ(d).

B∈Mn(R)
bij =1i|j
0
D= Diag(ϕ(1), . . . , ϕ(n)) ∈Mn(R)
∀n∈N∗, n =X
d|n
ϕ(d).
BTDB
S∈Mn(R)
∀(i, j)∈J1, nK2, sij =i∧j.
A A
A
{0}A A
A A
A
Q[i] = {a+ib ∈C|a, b ∈Q}
d∈N√d /∈N
Q[√d] = {a+b√d∈R|(a, b)∈Q2}
K
A={F∈K(X)|F(X) = F(1/X)}.
AK(X)
KY
x∈K∗
x
Q
C
fR
f:R→R
x∈Qf(x) = x
f
f
A=
a b c
c a b
b c a
∈M3(R)
(a, b, c)∈R3 .
AM3(R)
A

R[X]
A= Vect(Xn|n>2).
AR[X]
AR[X]
K
A={F∈K(X)|deg(F)60}.
AK(X)
A
A
C0(R,R)
A
R
A
a∈A P ∈R2[X]
P(a)=0
AC
A= a b
−¯ b¯a∈M2(C)
(a, b)∈C2.
ARM2(C)
A
AMn(K)
AMn(K)
s:A→K
∀M∈Mn(K), s(M) =
n
X
k=1
m1k.
s
AKa∈A
fa:A→A, x 7→ ax.
fa
f:A7→ L(A), a 7→ fa
A n A
Mn(K)
AKa∈A
fa:A→A, x 7→ ax.
f
A a ∈A
a0
∀a∈A×,∃P∈K[X], a−1=P(a).
B A B×=A×∩B