Résumé d`arithmétique des entiers et des polynomes.

Z={...,−3,−2,−1,0,1,2,3, . . .}
a b b a
k a =kb
p≥2 1
p N =p!+1 N p 1
p
a b ≥1 (q, r)
a=bq +r0≤r < b
a≥1b≥1m≥1
c≥1c a b c m
a≥1b≥1d≥1
c≥1d a b a
b
s t
n≥2
n=pα1
1...pαk
ket n=qβ1
1...qβi
i,
k≥1, ,i ≥1p1< p2< . . . < pkq1< ...qi
α1...αkβ1...βi
k=i1≤j≤k pj=qjαj=βj
a b a b
a b
(2573,23112) = 2373112a.b = (a, b)×(a, b)

A=R C A
a0, a1,...an. P (X) = a0+a1X+...+anXnX an6= 0
P n
A→A
xAP(x)X x
KAK[X]B
K[X]
X2X1X2
X22X X −1
−X1
−X−2
3
X2+X+ 1 = (X−1)(X+2)+3
K K =R C P(X) =
a0+a1X+...+anXnP z0
0z=z0P
X−z0P(X)=(X−z0)Q(X) + R(X)R(X)
X−z0R(X)r0P(X)=(X−z0)Q(X) + r0
X=z0.0 = P(z0) = r0P(X)=(X−z0)Q(X)P
z0X−z0
≥1
n n 1
1
1 0
1
P(X) = a0+a1X+...+anXn
a0, a1. . . anCP(X) = an(X−λ1)(X−λ2)...(X−λn)
λiλ=a+ib b 6= 0 λ=λi
i∈ {1,...,n}P(λ) = 0
0 = P(λ) = a0+a1λ+...+anλn
0=0=a0+a1λ+...+anλn=a0+a1.λ +...+an.λn=a0+a1λ+...+an(λ)n.
λ=a+ib λ =a−ib P λ2λ2, . . . λ =λi
λ(X−λ)(X−λ) = X2−(λ+λ)X+λλ =X2−2aX +(a2+b2)
∆=4a2−4(a2+b2) = −4b2
P1
1R[X]
1
N
{1,2,3,...N},2
3. . . P artieEnti`ere(√N+ 1)

1NR[X]C[X]
1
1
A(X)=(X−1)3B(X) = (X+ 1) P Q
1 = AP +BQ Y =X+ 1 ˜ A(Y)=(Y−2)3˜ B(Y) = Y
˜
A(Y) = Y3−3Y2.2+3.Y.22+ 1 = Y3−6Y2+ 12Y+ 1 Y
˜
A(Y) = (Y2−6Y+ 12)Y+ 1
1 = ˜ A(Y)−(Y2−6Y+ 12) ˜ B(Y)
X1 = A(X)−((X+ 1)2−6(X+ 1) + 12)B(X)
A(X)=(X−λ1)α1B(X)=Πk=s
k=2(X−λk)αkλi
P Q
1 = A(X)P(X) + B(X)Q(X).