Polynômes et fractions rationnelles

Z
k[X]k
k[X]
k
aX2+bX +c
anXn+an−1Xn−1+· · ·+a1X+a0aik X
anXn+an−1Xn−1+· · · +a1X+a0
(ai)i>0
ai= 0 i>n (ai)i>0
A(ai)i>0k k
k×
(ai)×(bi) = (ci)ci
d´ef
=X
r+s=i
arbsi>0
A×1A
d´ef
= (1,0,0, . . . )
k→A λ λ1A

A(ai)×(bi)A
×1A+
A k kNk
(ai) (bi)A(ci)=(ai)×(bi)
A ar= 0 r > d bs= 0 s > e arbs6= 0 r6d
s6e r +s6d+e i > d +e(r, s)
r+s=i arbs= 0 ci= 0 (ci)×
A
i((ai)×(bi)) ×(ci)
(ai)×((bi)×(ci))
X
r+s+t=i
arbsct.
×ci=Parbs(ai) (bi)
1A(δi,0)
1A×+
ci=Parbs(ai) (bi) (bi) (ai)
A×
A
k
X aX2+bX +c
(0,1,0,0, . . . )
X(0,1,0,0, . . . )Xk= (0,...,0,1,0, . . . )
k>1k Xk= (ai)ai=δi,k
P= (ai)ai= 0 i > n P =anXn+an−1Xn−1+· · ·+a1X+a0
a0a01A
k k = 1
Xk= (δi,k)i>0i Xk+1 =Xk×X δr,kδs,1
r, s r +s=i r =k s = 1
Xk+1 i=r+s=k+ 1 1
k+ 1
aiXi
aii
X k[X]
X k k
k[X]λ=λ1k[X]
aiXi
P=anXn+an−1Xn−1+· · · +a1X+a0X
P6= 0 Pdeg(P)i ai6= 0
P= 0 deg(P) = −∞

P6= 0 Pval(P)i ai6= 0
P= 0 val(P) = +∞
deg(P) = n6=−∞ anXnP an
P
P=PaiXiP=P∞
i=0 aiXi
ai
P=anXn+an−1Xn−1+· · · +a1X+a0
an6= 0 P n P
0
P Q
deg(P Q) = deg(P) + deg(Q)
deg(P+Q)6max(deg(P),deg(Q)) deg(P)6= deg(Q)
val(P Q) = val(P) + val(Q)
val(P+Q)>min(val(P),val(Q)) val(P)6= val(Q)
P=PaiXiQ=PbjXjp= deg(P)q= deg(Q)
P Q i P Q
apbqi=p+q i > p +qdeg(P Q) = p+q P Q
−∞
P Q i > max(p, q)ai+bi= 0+0 = 0 deg(P+Q)6max(p, q)
p6=q p > q p P +Q ap
P Q P = 0 max(deg(P),deg(Q)) = deg(Q)
k[X]P Q k[X]
P Q
P, Q ap, bqk
P Q apbqP Q 6= 0 
k[X]
P k[X]Q P Q = 1 P Q
deg(P) + deg(Q) = deg(1) = 0 deg(P) = deg(Q) = 0 P Q

k[X]
A ab = 0 ⇒a= 0 b= 0
A p, a, b A
a b a |b c ∈A b =ac a
b b a
a(a)a
a b a ∼b λ ∈A a =λb
p a, b p =ab
a b
(a)a
a|b b ∈(a)
a|b b |a a b ∼
k[X]k[X]\ {0}
(a) = A a A
p=ab a b
p
k[X]
A B B 6= 0
(Q, R)
A=BQ +R
deg(R)<deg(B)
A B A B
Q R
(Q, R) deg(A)<deg(B)
Q= 0 R=Adeg(A)>deg(B)A=anXn+· · · +a0
B=bmXm+· · ·+b0bm6= 0 Q
i∈ {0, . . . , n −m+ 1}Qideg(A−BQi)6n−i
i= 0 Q0= 0 Q0, . . . , Qii6n−m
A−BQi=cn−iXn−i+· · · +c0deg(A−BQi)>n−i>m
Qi+1 =Qi+cn−ib−1
mXn−i−m
A−BQi+1 =A−BQi−Bcn−ib−1
mXn−i−m
= (cn−iXn−i+· · · +c0)−(bmXm+· · · +b0)cn−ib−1
mXn−i−m
=cn−iXn−i−bmcn−ib−1
mXn−i
| {z }
=0
+ ( 6n−i−1)

deg(A−BQi+1)6n−(i+ 1)
i=n−m+ 1 Q=Qn−m+1 deg(A−BQ)6m−1
R=A−BQ
(Q, R)
(Q0, R0)A=BQ +R=BQ0+R0B(Q0−Q) = R−R0R−R0
Bdeg(R−R0)6max(deg(R),deg(R0)) 6deg(B)−1
R−R0= 0 B(Q0−Q) = 0 Q0−Q= 0 
Z
P, Q, . . .
k[X]
I k[X]I={0}0
P I P
I F ∈I P F P
F=P Q +Rdeg(R)<deg(P)P F I
R=F−P Q ∈I R < deg(P)P
I R = 0 P F 
P, A, B P P |AB P |A
P|B
P-A P |B I = (A, P )A
P I = (D)D
A∈I P ∈I A0, P 0A=DA0P=DP 0P-A
D6∼ P P D ∼1
D I = (D) = A1∈I
U, V 1 = UA +V P Q AB =P Q
B=BU A +BV P =U P Q +BV P = (UQ +BV )P P 
F
F=λP α1
1. . . P αr
r
λ∈k∗Piαi>1
F=λP α1
1. . . P αr
r=µQβ1
1. . . Qβs
s
λ=µ r =s σ ∈SrPi=Qσ(i)αi=βσ(i)