TD d`algèbre générale - Jean

n A Nf
gR
n n
n A A N
f f g
x7→ x2x7→ cos2(x)+2
n n
n n
ABC A [BC]
a|b a b P
∀n∈N,(6|n∧4|n) =⇒24|n
∀n∈N,(6|n∧n|40) =⇒n∈ P
∀p∈ P,∀a∈N,∀b∈N,(p|a p|b=⇒p|a+b
2)
∀n∈N\ {0,1},∃p∈ P,∃q∈ P,2n=p+q
∀n∈N,2n−1∈ P =⇒n∈ P
∀x∈R, x2≥x
∀x∈R,∃!y∈R, xy = 1

∀x∈R,∃a∈Z∗,∃b∈Z,∃c∈Z, ax2+bx +c= 0
x
3
√2
P
R
f D y = 2x+ 3
f D
P
E E ={f:R→R|fR}
∀f∈E, f ⇔f0
∀f∈E, f ⇔f0
∃!f∈E f0=f
∀f∈E, ∀g∈E, f =⇒g◦f
∀f∈E, ∀g∈E, f =⇒f◦g
h(x) = f(−x)h0(x) = −f0(−x)
f◦g g f f ◦g(x) = f(g(x))
n
∃k∈N, k ≥2k2|n
∀k∈N, k|n=⇒k2|n
∀k∈ P, k|n=⇒k2|n
∃k∈ P,∃j∈ P, n =kj
∀k∈N,∃j∈N, n|k n|(k+j)
∃j∈N,∀k∈N, n|k n|(k+j)
log10(2)
log10
x7→ 10x

∀n∈N,7 32n+1 + 2n+2
∀n∈N,3 4n+ 5
∀x∈R+,∀n∈N,(1 + x)n≥1 + nx
n∈N∗k∈ {0, . . . , n}n
k n k=n! k!(n−k)!
0! = 1
∀n∈N∗,∀k∈ {0, . . . , n −1},n+1
k+1=n k+n k+1
n= 1,...,10
n
p∈ P ∀k∈ {1, . . . , p −1}, p p
kp
p∈ P a∈Np ap−a
a

a= 147 b= 63
a b
PGCD p a b
u v au +bv =p
a= 1 111 111 111 b= 123456789
30233023 Z/nZn= 2,3,4,5
Z/3ZX2+X+ 1 X2+ 1 X3−X
X3+ 2X+ 1
¯n2¯n∈Z/13Z
Z/13Zx2+x+¯ 7 = ¯ 0
Z/12Zx2+¯ 4x+¯ 3 = 0
x y
z
x2+y2= 4z+ 3, x2−2y6= 17, x2+y2= 9z+ 6.
Z/4Z Z/7Z Z/3Z
(x, y)∈Z22x+ 3y= 7
15x−6y= 14