Feuille de TD nº4

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A a ∈A a
x∈A x ·a= 1 x0∈A a ·x0= 1)
(1) a x =x0a
(2) A×A(A×,·)
A
(3) (Z/6Z)×(Z/7Z)×
(4) (Z/nZ)×
Z[i] := {a+ib :a, b ∈Z}.
(1) Z[i]C
(2) Z[i]
A x ∈A∀y∈A\ {0}x·y6= 0
(1) φ(y) = x·y x
(2) x
(3) Z/10Zx6= 0
y6= 0 x·y= 0 Z/10Z
R2?
(x, y)?(x0, y0) := (xx0, xy0+x0y).
(1) A:= (R2,+, ?)
A
(2)
f:R[X]→A
P7→ (P(0), P 0(0))
f
f
f

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n∈N∗Mn(R)
n×n
In=
1 0
0 1
.
n= 2 J=0−1
1 0 K M2(R)
K:= {aI2+bJ :a, b ∈R}.
(1) I+J2= 0
(2) (K,+,×)
(3)
A
x∈A x2= 0
(1) x∈A x +x= 0 A
(2) n≥1 (Z/2Z)n
(3) EAE
Z/2ZA
(1)
(2)
(3)
n≥1Z/nZ
n
(1) n= 9 I:= {0,3,6}Z/9Z
(2)
(1) D:= {P∈R[X] : P0(0) = 0}D
R[X]R[X]
(2) E:= {P∈R[X] : P(0) = P0(0) = 0}E
R[X]R[X]

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0
A a ∈A(a) := aA
(1) a∈A(a)A
a b A u
A a =bu
(2) a b (a) = (b)
(3) A(a)=(b)a b