Fiche 4 sur les anneaux et corps

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[•]
Z+iZ={a+ib|a∈Z, b ∈Z}
Z+iZ(C,+,·)
x∈Z+iZN(x) = |x|2|·| C
x∈Z+iZ Z +iZN(x) = 1
Z+iZ
[•]
A
x∈A xA ={y∈A|∃a∈A:y=xa}
A x
xA =A x A A
{0}A A
[•]
A I, J A
I∩J A I J
I+J={z∈A|∃x∈I, ∃y∈J:z=x+y}A
I J
I∩J=I+J
A a, b ∈A d m a b
a b A bA ⊂aA
aA +bA =dA aA ∩bA =mA
Z
xZ+yZxZ∩yZx= 23 y= 41
x= 121 y= 737 x= 72 y= 84
[•]
K={a+b√2|a∈Q, b ∈Q}R
(A, +,·)A C ={x∈A|∀y∈A, xy =yx}
C A
R⊕ ⊗
∀x∈R,∀y∈R, x ⊕y=x+y−1x⊗y=x+y−xy.
(R,⊕,⊗)
M={aI2+bJ|a, b ∈R}I2=1 0
0 1J=0 2 1 0
J2a, b ∈RaI2+bJ = 0 a=b= 0
M2(R)M
M

[•]
(A, +,×)x∈A n ∈N∗xn= 0
x1−x
x y xy x +y
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Z/5Z Z/12Z
Z/12Z5x= 5 6 x= 6
Z/12Z10 nn∈N∗
[•]
xZ/nZx n
p
Z/pZ
Z/pZ
[•]
n1n2≥2u
v un1+vn2= 1
a1a2aZa≡a1vn2+a2un1[n1n2]x∈Z
(x≡a1[n1]x≡a2[n2]) ⇐⇒ x≡a[n1n2].
x≡1 [17]
x≡2 [28]
x≡3 [31]
.
φZ/(n1n2)Z Z/n1Z×Z/n2Z¯x
( ˙x, ¨x)Z/24Z Z/8Z×Z/3Z
(A, +,×)A×A∗
R(p, q)R(p0, q0)⇐⇒ pq0=p0q,
(p, q)+(p0, q0)=(pq0+p0q, qq0),(p, q).(p0, q0) = (pp0, qq0).
RR
K= (A×A∗)/Rp
q(p, q)
K(K,+, .)
A
Φ : A→KΦ(p) = p
1
A=ZA

(A, +,×)A
Af:R−→ AA
+×f, g ∈ A
f+g:R−→ A
x7−→ f(x) + g(x)
f×g:R−→ A
x7−→ f(x)×g(x)
(A,+,×) 0A:R−→ A
x7−→ 0A
1A:R−→ A
x7−→ 1A
f∈ A fAx∈Rf(x)
A
f∈ A f
x∈Rf(x)=0A
A
I A If:R−→ I
AB A Bf:R−→ B
A
Z/20Z
Z/20Z
Z/4Z×Z/6Z