118: Exemples d`utilisation de la notion de dimension

E F k
L K L
K−ev L
L
kZ/pZ
|k|=pn
k⊂K⊂L[K:k][L:K] = [L:k]
L L =K(α)α
T7→ P(α)
µα
α
K[α] = K(α)
dimKK[α]<∞
P K[X]
p(2) QX2−2Q[p(2)]
3p(7) + p(2) : X6−6X4−14X3+ 12X2−84X+ 41
i+j:X4+ 2X3+ 5X2+ 4X+ 1
j+p(3) : X4+ 2X3−3X2−4X+ 13
jp(2) : X4+ 2X2+ 4
K

P L
L=P(α)P(α) = 0 L[X]
K
Fq
nFp
Fpn
α
αpi
pnpFpdd
nFqFqn
A M
A M
A
A A
A1,...An
A0= (0,0),(1,0)
M∈An
Ai=Ai−1∪MiMiAi−1
x(x, 0)
Q
E F E F
F

f(ei)ei
B T r(MB(ϕ) = 0
1,2, . . . , n
Mn,m(R)A B
P∈GLn(K)Q∈GLm(K)B=P AQ f ∈L(E, F )
g∈L(E, F )u∈Gl(E)v∈Gl(F)f=vgu
(B1, B2) (B0
1, B0 2)
MB1,B0
1(f) = MB2,B0 2(g)
A B P ∈Gln(R)A=tP AP
q0=qα
B B0MB(q) =
M0
B(q)
dimF +dimF −
F+F−

E∗fidim T
Ker(fi)
=n−rg((fi))
r n −r
(fi)T
Ker(fi)
=F
x0
n−r(fi)TKer(fi) + x0=F
ϕ1(x) =
b1, ϕ2(x) = b2, ϕ3(x) = b3
ϕi
ϕ3=λϕ1+µϕ2b36=λb1+µb2
VRna∈V d V
a d C1U
aRnd F (U∩V) = V0∩F(U)
V0=Rd× {0} ⊂ RnV d RnV
d
CkC∞
0R2
URnf1, . . . , fpURC1
V x x ∈U f1(x1,...xn) = 0 . . . fp(x1,...xn) = 0
vRn
x2−y2= 0
x3+y3−3xy = 0

a d
aRn−p
U∩M=g−1(0)
ARp(a1, . . . , ap)
C∞U∩M
pMn(R)
p(2n−p)
V a d a
d a V
a
P∂jfi(a)vj= 0
SLn(R)n2−1
SLn(C) 2(n2−1)
On, SOn(R)tM=−M
n(n−1)/2
Un(C)t¯ M=−M n2