Quelques applications des algèbres de matrices à la

∗
K E K A L(E)
E A {0}E
K E K L(E)
E A L(E)
A E − {0}E∗− {0}
∀x∈E− {0} {a(x), a ∈A}=E
∀ϕ∈E∗− {0} {ϕ◦a, a ∈A}=E∗.
A A =L(E)
K A
K A K A {0}A
K A K A
E K A L(E)
J A
A J ρ :A→ L (J)
∗

E K dimKE
k K A =k⊗KE a ∈A p ∈N
(λi)∈kp(xi)∈Ep
a=
p
X
i=1
λi⊗xi.
p(a)p
a∈A p (a) = 1 a
a∈A a =Pp(a)
i=1 λi⊗xi(λi) (xi)K k E
0 = Pp
i=1 λi⊗xi(λi)K k (xi)E xi
λi
E K A k
dimKE=dimkA.
I A a ∈I p (a) = 1 I=A
k
Q
a b
p∈N
p≡amod b.
p∈NΦp= 1 + X+... +Xp−1p
ΦpQ
(Z/pZ)∗
q∈Np(Z/pZ)∗Φp
Fq
KQ FpLΦpK L =K[u]
uΦpL K σ L σ (u) = uq
σ p −1
L σ K
σi(u)06i6p−2L K
α∈L
N(α) =
p−2
Y
i=0
σi(α).

N(α)∈K N (α) = 0 ⇐⇒ α= 0
P∈Z[X0...Xp−2]K
si α=
p−2
X
i=0
xiσi(u),avec xi∈K, alors N(α) = P(x0...xp−2)
K=Q
(xi)06i6p−2∈Zp−1α=Pp−2
i=0 xiσi(u)∈L N (α)∈Z
N(α)≡0 mod q⇒ ∀i∈ {0...p −2}xi≡0 mod q.
Q(p−1)2
K=QMp−1(L)Q
M:L→ Mp−1(L)
α7→ M(α) = diag α, σ (α), ..., σp−2(α).
A=
0q
1 0
0
1 0
.
E=(p−2
X
i=0
M(αi)Ai,(αi)∈Lp−1)⊂ Mp−1(L).
M
Ap−1=q Ip−1
∀α∈L M(α)A=A M(σ(α))
EQMp−1(L) (p−1)2Q
det p−2
X
i=0
M(αi)Ai!≡N(α0) mod q,
∀N∈Edet N= 0 ⇒N= 0.
E
p∈N(p−1)2
Q
nQ
n2Q

F E n
p
n p −1r=p−1
n
n r
p p ≡(n+ 1) mod n2
M L σnτQM
σrv=Pr−1
j=0 σj(u)
τ n
M τ Q
τi(v)06i6n−1MQ
B=Ar
Bn=q Ip−1
∀β∈M M(β)B=B M(τβ)
F=(n−1
X
i=0
M(βi)Bi,(βi)∈Mn)⊂E⊂ Mp−1(L).
Qn2
Q