Feuille de T.D. 4 - Structures algébriques

Z
•
?Rx?y=x+y−xy
(R, ?)
(R, ?)
x∈Rn∈N\{0}x(n)=x ? x ? ··· ? x
x?y= 1 + (1 −x)(y−1)
? e G
x ? x
∀y, y0∈G, x ? y =x?y0=⇒y=y0.
(G, ?)e x?x =e x ∈G G
Unn−(C,×)
(G, ?)GR
xRy⇐⇒ ∃a∈G, y =a?x?a−1.
RG e
Z[i] = {a+ib, a, b ∈Z}
(Z[i],+,×)C
(C,+) 1
•Z
5nn∈N) 5789
7a= 247349
7n+ 1 n
n8
n∈N3n+3 −44n+2 11
A B
3A∩B A ∪B
Z
3x≡7[9]
4t≡2[5]
x, y z x3+y3=z3
x y z 3

a a3≡1[9] a3≡ −1[9]
Z262x+ 43y= 3 (E)
62 43
62 43
(x0, y0) (E)
(E)
1665x+1035y= 45
n∈Z5n−9 2n−6
n2n+ 1 n∈Z
a, b c
a∧b= 1 a∧c= 1 ⇐⇒ a∧bc = 1
a∧c= 1 =⇒a∧(bc) = a∧b
a b 0< a < b
(a+b)∧(a∨b) = a∧b.
a b a+b= 144
a∨b= 420.
P(X) = X2+aX +b a, b
α P (X)=0 α
α α =p/q p ∧q= 1
n√n P
√n1.
m∈N\{0}am+ 1 m
m= 2nn∈N
xl+ 1 = (x+ 1)(xl−1−xl−2+··· + 1) x∈Rl∈N
a n a, n ≥2
an−1a= 2
Mn= 2n−1n
M11
p
pp
kk∈ {1,2, . . . , p −1}
a b (a+b)p−ap−bpp
(a+ 1)p−(ap+ 1) p
a a ap−a p
p a p|ap−1−1