10 Polynômes à une indéterminée à coe cients réels ou complexes

K
R C
(n, k)∈N2,
δn,k =1k=n
0k̸=n
K[X].
KNa= (ak)k∈NK.
a+b= (ak+bk)k∈Nλa = (λak)k∈N
a, b KNλK K
KNa= (ak)k∈N, b = (bk)k∈NKN,
ab = (ck)k∈N, ck
∀k∈N, ck=
k
j=0
ajbk−j=
k
j=0
ak−jbj
KN
P0= (δ0,k)k∈N.
KNK
KN
ab = (akbk)k∈N

a= (ak)k∈N∈KN
Supp (a) = {k∈N|ak̸= 0}
P= (ak)k∈N∈KN
m∈Nak= 0 k≥m.
0
K(N)
P= (ak)k∈Nk∈N
ak̸= 0
deg (P) = max {k∈N|ak̸= 0}
val (P) = min {k∈N|ak̸= 0}
deg (P)Pval (P)P.
deg (0) = −∞ val (0) = +∞
P n ≥0
P= (a0,···, an,0,··· ,0,···)
an̸= 0 P.
K(N)KN(Pn)n∈N
∀n∈N, Pn= (δn,k)k∈N= (0,··· ,0,1,0,··· ,0,···)
K(N).
0P0K(N)
P, Q K(N)λK, P +λQ P Q K(N).K(N)
KN.
P= (ak)k∈Nn,
P=
n
k=0
akPk
(Pn)n∈NK(N).
n,
n
k=0
akPk= 0 (a0,··· , an,0,··· ,0,···) = 0,
ak= 0 k0n. (Pn)n∈N
K(N).
a0∈K7→ (a0,0,··· ,0,···) = a0P0∈K(N)
K K(N),K K(N)
0. a0P0, a0∈K,
P0K(N)1.

K[X].
P, Q K(N),
deg (P+Q)≤max (deg (P),deg (Q))
deg (P)̸= deg (Q)
val (P+Q)≥val (deg (P),deg (Q))
val (P)̸= val (Q).
P, Q K(N),
deg (P Q) = deg (P) + deg (Q)
val (P Q) = val (P) + val (Q)
P= 0,deg (P) = −∞ P+Q=Q, P Q = 0,
deg (P+Q) = deg (Q) = max (deg (P),deg (Q))
deg (P Q) = −∞ = deg (P) + deg (Q)
P Q Q = 0.
P= (a0,··· , an,0,··· ,0,···)Q= (b0,··· , bm,0,··· ,0,···)K(N)
n= deg (P)≤m= deg (Q),
P+Q= (a0+b0,···, am+bm,0,··· ,0,···)
am+bm̸= 0 n < m, P +Q m,
n < m,
P Q = (c0,··· , cn+m,0,··· ,0,···)
cn+m=anbm̸= 0, P Q n +m.
K(N)
P, Q P Q nm ̸=−∞, P Q ̸= 0
1 = P0= (1,0,···,0,···)K(N)X=
P1= (0,1,0,···,0,···), n ≥1, Xn=Pn. n = 1
n≥1,
Xn+1 =X·Xn=P1Pn= (δ1,k) (δn,k) = (ck)k∈N
cn+1 =
n+1
j=0
δ1,jδn,n+1−j=δn,n = 1
k̸=n+ 1
ck=
k
j=0
δ1,jδn,k−j=δn,k−1= 0
ck=δn+1,k k Xn+1 =Pn+1.

n P =
n
k=0
akXk,
X0= 1.
P(X) =
n
k=0
akXk.
XK[X]
K.
λXnλ∈Kn∈Nn= 0,
n, Kn[X]K[X]
n.
Kn[X]K[X].
Kn[X]n+ 1,Xk0≤k≤n
Kn[X]
P(X) =
n
k=0
akXkn≥0an= 1.
K[X]
P=
n
k=0
akXkQK[X].
P Q
P◦Q=
n
k=0
akQk
Q0= 1 Q.
K[X]K
P(Q)P◦Q. Q =X, P (X)
P(X)
Q=X−α, α ∈K,
P(X−α) =
n
k=0
ak(X−α)k
P.
P, Q K[X],
deg (P◦Q) = deg (P) deg (Q)
P=
n
k=0
akXkn≥0Q=
m
j=0
bjXj, Qk
mk, anQnnm akQk
(n−1) m < nm k 0n−1.

PK[X]. P
P′
P′(X) = 
0P= 0
p
k=1
kakXk−1P(X) =
p
k=0
akXkp≥1
Xk′=kXk−1k≥1P
p≥1, P ′p−1,K
pap−1̸= 0 ap̸= 0
Kp≥2,(Xp)′=pXp−1= 0 Xp
P′= 0 P P
p≥1P′p−1
P7→ P′K[X]K[X],Kp[X]
Kp−1[X]p≥1P, Q K[X],
(P Q)′=P′Q+P Q′,(P◦Q)′= (P′◦Q)Q′
P=a0∈KQ∈K[X], P ′= 0 (P Q)′=a0Q′=P′Q+P Q′.
P Q P ∈K[X]
Q
P(X) =
p
k=0
akXkp≥1K[X]Q(X) = Xqq∈N∗,
(P Q)′=
p
k=0
ak(k+q)Xk+q−1=Xq
p
k=0
kakXk−1+qX−1
p
k=0
akXk
=P′Q+P Q′
(P Q)′=P′Q+P Q′P, Q.
k≥1Qk′=kQk−1Q′
Q. k = 1 k≥1,
Qk+1′=QQk′=QQk′+Q′Qk=kQQk−1Q′+Q′Qk= (k+ 1) QkQ′
P=a0∈KQ∈K[X],
(P◦Q)′= (a0)′= 0 = (P′◦Q)Q′
P(X) =
p
k=0
akXkp≥1Q∈K[X],
(P◦Q)′=
p
k=0
akQk′=p
k=0
kakQk−1Q′= (P′◦Q)Q′