Agrégation externe de Mathématiques

E K mu
u∈ L(E)χu
u∈ L(E)F u v =u|Fmv/mu
χv/χuF1F2u ui=u|FiE=F1+F2
mu=ppcm(mu1, mu2)E=F1⊕F2χu=χu1×χu2
u∈ L(E)mu=Qr
i=1 Qαi
imu
k[X]
E=⊕r
i=1 ker Qαi
i(u)
j>αi
{0}(ker Qi(u)(ker Q2
i(u)(· · · (ker Qαi
i(u) = ker Qi(u)j,
u u|ker Qβ
i(u)Qβ
i
β6αi
K=Ru E
u
u E E
u
M∈ Mn(K)L⊃K K
M L[X]K[X]M
L[X]K[X]K L mL
M
muχu
u∈ L(E)P u u
P χuk[X]mu/χu/mn
u

M∈ Mn(k)u E Mn(L)L
χMmu/χu/mn
uu∈ L(E)
χumuk[X]χu=Qd
i=1 Qβi
i
χu(αi)1≤i≤d1≤αi≤βi
mu=Qd
i=1 Qαi
i
Ei= Ker Qβi
i(u)u E =⊕d
i=1Ei
Eiu Ei= Ker Qαi(u)ui=u|Eiχui=Qβi
i(u)
mui=Qαi
i(u) dim Ei=βi×deg Qi
F E u F =⊕d
i=1F∩Ei¯u=u|F
IF=i∈ {1,· · · , d}/F ∩Ei6={0}χ¯u=Qi∈IFQβ0
i
im¯u=Qi∈IFQα0
i
i
1≤α0
i≤αi1≤β0 i≤βii∈IF
u u
u
R3
1−1−1
−1 1 −1
−1−1 1
.
M∈Gln(C)N∈N∗MNM
M M ∈Gln(R)
u∈ L(E)x∈E Ex={P(u)(x)/P ∈K[X]}Px
K[X]P(u)(x) = 0 Px/P
ExE u x
Px/muExdx= deg(Px)u|Ex
(x, u(x),· · · , udx−1(x)) Px/χu
Ex∩Ey={0}Px+y=ppcm(Px, Py)PxPy
Ex+y=Ex⊕Ey
P∈K[X]P/mux∈E Px=P
P
M∈ Mn(C) 0
{P MP −1/P ∈Gln(C)}M M
M M
ECn(e1,· · · , en)E
σ∈ SnE u(ei) = eσ(i)uσ
uσσ
σ