Anneaux Exercice 1. (NEW) Exercice 5. Thm de Gauss

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A K = (A)B K
A(B) = K
E A =P(E)
(A, ∆,∩)
I A (∀X∈I, ∀Y⊂X, Y ∈I
∀X, Y ∈I, X ∪Y∈I.
I=P(E0)E0⊂E
E I ={E} P(E0)
A{0}A A
I A ∀x, y ∈A, xy ∈I⇒x∈I y ∈I
Z
A A A
A a, b ∈A(a b b ∈aA
a b aA +bA =A.
a b a bc a c
A A 1 (A, +)
A n
∀x∈A, nx = 0
A n
A x 7−→ xn
A∀x∈A, x2=x
A
∀x∈A, x +x= 0 A
x, y ∈A x ≤y⇐⇒ ∃ a∈A x =ay
A a ∈A a n ∈Nan= 0
Z/36Z
A
a1−a1=1n−an

a b b +a
A I A
√I={x∈A∃n∈Nxn∈I}I
√I A
p√I=√I
√I∩J=√I∩√J√I+J⊃√I+√J
A=ZI= 3648Z√I
A I, J A
IJ ={a1b1+··· +anbnai∈I, bi∈J}
IJ A
I(J+K) = IJ +IK
I+J=A IJ =I∩J
A=ZI=nZJ=pZIJ
A
RA
I0I A
J A x ∼y⇐⇒ x−y∈J∼
+×
Z2
d∈NAd={(x, y)∈Z2x≡y[d]}x=y d = 0
AdZ2
Z2
IZ2(I1={x∈Z(x, 0) ∈I}
I2={y∈Z(0, y)∈I}.
I1I2ZI=I1×I2
I
KE
KE A =KEe∈E Ie={f∈A f(e) =
0}, χe:x7−→ 1six =e
0six 6=e.
Ie
f∈A f =P
e∈E
f(e)χe
I A F ={e∈E∃f∈I f(e)6= 0}
IP
e∈F
χe
A=f:R→Rf(x) = a0+
n
P
k=1
akcos(kx), n ∈N, ai∈R
ARR
f∈A f = 0 akZ2π
t=0
f(t) cos(nt)t
A

A(In)A I =S
n∈N
In
I A
A n0∈NI=In0
RR
G A ={f:G→G}
(A, +,◦)
G=Z/nZn≥2Aϕ:G−→ G
x7−→ kx
k∈G A Z/nZ
A={m/n ∈Qn}
AQ
A
A2kk∈N
Zn>Z
Zn>Z
A={ }
A
A > Z
I={ }
A={a+bi a, b ∈Z}
AC
u, v ∈A v 6= 0 q, r ∈A u =qv +r|r|<|v|
A
A
A
A In=
xnA x ∈A
F4
Kx∈K∗O(x)x n
K∗
a, b ∈N∗a0, b0∈N∗a0|a b0|b a0∧b0= 1 a0b0=a∨b
x, y ∈K∗a b u, v O(xuyv) = a∨b
z∈K∗n
n= (K∗)K∗

Kn a, b ∈Nab =n−1
fa:K∗−→ K∗
x7−→ xafaNa= ( fa)
Na≤a
(fa)⊂fbNa=a Nb=b
ϕ a n−1
K∗a ϕ(a)K∗
K(K, +,×) (K\ {0},×)
n∈N∗PnnCΦ1(X) =
X−1 Φn(X) = Q
ζ∈Pn
(X−ζ) Φnn φ(n)φ
(∀n∈N∗)Xn−1 = Q
d|n
Φd(X) Φn(X)
Z
Φn(X)n≤16
p α ∈N∗Φpα(X) =
p−1
P
k=0
Xkpα−1
Φn
d < n d n Xd−1Xn−1Z[X] Φn(X)
Xn−1Xn−1
Xd−1Z[X]
K Z(K)q
Z(K)
K Z(K)n
K q n
a∈K\ {0}Ca={x∈K|ax =xa}
CaZ(K)d
n K Ca
K∗
qn−1 = q−1 +
k
X
i=1
qn−1
qdi−1i, di|n
Φn(q)q−1
|Φn(q)|n= 1

A{0}a∈A\{0}a×a∈(a2)a2
a a
x+y= (x+y)2=x2+y2+xy +yx =x+y+xy +yx ⇒xy +yx = 0
y= 1 x+x= 0 ⇒1 = −1
y xy =−yx =yx
x=ay xy =ay2=ay =x
(x≤y) (y≤x)⇒xy =x=y
A=Z[X]I= (X)J= (X+ 4)
114Z
f(x1, . . . , xn) = a1x1+··· +anxn
f⇒aiaj= 0 i6=j
f(1,...,1) = 1 ⇒ai1 0
f
±1,±i
1 + i= 0 ×2 + (1 + i) = 1 ×2+(i−1)
K={0,1, a, b} {1, a, b} ⇒ b=a2a3= 1
+ 0 1 a a2
0 0 1 a a2
1 1 0 a2a
a a a20 1
a2a2a1 0
×1a a2
1 1 a a2
a a a21
a2a21a