129:algèbre des polynômes d`endomorphismes en

ϕu:K[X]→ L(E)P P (u)
K K[u]L(E)K[u] =
P(u)|P∈K[X]
K[u]
K[u]'K[X]/Ker(ϕu)Ker(ϕu) 0
K
u K[u]
Ker(ϕu) ΠuK[X]
ΠuΠuu
uL(E)
K
µu)61 + rg(u)
K=Cp0
FΠuF|ΠuG
FΠu=ppcm(ΠuFΠuG)
K[u]'K[X]/(P1K[X]) . . .×
K[X]/PrK[X]P1, . . . , Pr
K[u]⇔µuK[X]↔
u7→ Πu
(X−1)(X−1−1/k) (X−1)2
X−1

K[u]L(E)
Panun
u Pu
u Pu(u) = exp(u)
P(u)u
P A λ P (λ)=0
P u(x) = λx P (u)(x) = P(λ)x DoncSp(P(A)) ⊂
P(Sp(A))
X(X−1)(X−2) p2
λ u ⇔Πu(λ)=0
Im(P(u)) + Im(Q(u)) = Im(pgcd(P, Q)(u)
Im(P(u)) ∩Im(Q(u)) = Im(ppcm(P, Q)(u)
Ker(P(u)) + Ker(Q(u)) = Im(ppcm(P, Q)(u)
Ker(P(u)) ∩Ker(Q(u)) = Im(pgcd(P, Q)(u)
P Q Ker(P(f)) Ker(Q(f))
Ker(P Q(f)f
k
χA=det(XIn−A)
A7→ χA
F G
χf=χfFχfG
u7→ χu
Πu|χuχu(u)=0
K[u]dim(E)

χu
Πu
A0//A AA
A0//C AC A C = 0 C A
k[U]'kll
χu
Πu
X
B
Πf
K
k[U]
k

⇒Sp(P(u)) = P(Sp(u))
upKer() = Ker(u2)⇒
⇒
Cexp(u)⇒R
u
u P
v P (v) = u
v
n
n
dimK[u] = n↔dimC(u) = n↔deg(µu) = n↔µu=χu↔C(A) = K[u]↔u
⇔u
u K[u] = C(u)u
Dim(C(u)) = Pn
1(2s−2i+ 1)deg(Pi) = Pi,h min(dim(Pi), dim(Pj)) Pi
K[u] = C(C(u))