Arithmétique Polynômes

a b 7a2−3b3= 6
(a, b) 3b3+ 6 0 7
b= 0,1,...,6 7 b30,1,1,6,1,6,6
3b3+ 6 6,2,2,3,2,3,3
a b
abc
b2−4ac
p=abc a >1
p2c>1p10
b2−4ac =d2d>0
F(X) = aX2+bX +c1
p=F(10)
b2−4ac =d22b
d
r>1s>1b−d= 2r b +d= 2s
ac =rs e = pgcd(a, r)>1f>1r0>1
a=ef r =er0er0s=rs =ac =efc r0s=fc
f s s0>1s=fs0
•ef =a
•er0+fs0=r+s=b
•r0s0a=r0s0ef =rs =ac r0s0=c
(eX +s0)(fX +r0) = efX2+ (er0+fs0)X+r0s0=aX2+bX +c
p= 100a+ 10b+c=F(10) = (10e+s0)(10f+r0)
k k[X]k
a, b k[X]Xa−1
Xb−1

a=bq +r r < b a b
Xa−1 = XbqXr−1 =(Xb−1 + 1)qXr−1
= q
X
i=0 q
i(Xb−1)i!Xr−1
= q
X
i=1 q
i(Xb−1)i−1!Xr(Xb−1) + (Xr−1).
deg(Xr−1)r < b = deg(Xb−1)
Q=Pq
i=1 q
i(Xb−1)i−1XrR=Xr−1
k1
2 3
k n >4
P λ ∈k
P X −λ P = (X−λ)Q P Q
deg(P) = 1 deg(P) = 1 P=aX +b a 6= 0
λ=−b/a
P n = 2 n= 3 P
>1n= 2 1
1n= 3 1 2 1
k=RP= (X2+ 1)24
4
kF2
k
2k[X]
3k[X]
4k[X]
k0 1
λ∈k P P (λ)=1
P P (0) = P(1) = 1
P=X2+aX +b

P(0) = P(1) = 1 b= 1 a= 1 P=X2+X+ 1
2
P=X3+aX2+bX +c
P(0) = P(1) = 1 c= 1 a+b= 1 P=X3+aX2+ (a+ 1)X+ 1
a= 0 a= 1 3
>4
4
4
2 1
2X2+X+ 1
4 (X2+X+ 1)2
4
{ } ={ } ∪ {(X2+X+ 1)2}.
P=X4+aX3+bX2+cX +d
P(0) = P(1) = 1 d= 1 a+b+c= 1
P=X4+aX3+bX2+(a+b+1)X+1 a b
k={0,1}4
a= 0 b= 1 (X2+X+1)23 4
•a=b= 0 X4+X+ 1
•a=b= 1 X4+X3+X2+X+ 1
•a= 1 b= 0 X4+X3+ 1