112 - Corps finis. Applications - IMJ-PRG

K
K φ :Z−→ K, n 7−→ n.1K
K(K)
ker φker φ
K(K) = p > 0KZ/pZ
Z/pZ(K) = pnn= [K:Z/pZ]
P∈K[X]n n
|K|=q K∗'Z/(q−1)Z
Xq−X=Qα∈K(X−α)
K p
pnn∈N∗r n
K prα∈K αpr=α
K p α K?
K=Z/pZ(α)n= [K:Z/pZ] (1,α,...,αn−1)K
Z/pZ
p n ∈N?q=pn
K q
Xq−XZ/pZ
KZ/pZ Fq
Sn∈N∗Fpn!
p

K p 6= 0 F:x7−→ xp
Z/pZK
K
K F K =Z/pZF
K p α ∈K r
Fpr αpr=α αpi06i<r
αFpQ
16i<r
(X−αpi)
q=pnpFq
n F :x7−→ xp
Fq
q p
p= 2 Fq
p6= 2 F?
qF?2 q
F?
qx7−→ x(q−1)/2
{±1}
p
a∈Z
a
p=
0p|a
1p-a a p
−1
a∈Z
a
p=a(p−1)/2mod pab p=a pb p
−1
p= (−1)(p−1)/22 p= (−1)(p2−1)/8
q6= 2 q
pp q= (−1)(p−1)(q−1)/4
4m+ 1
p V
Fpu∈(V)(u) = det u
p
F:x7−→ xpFpnp n ∈N?(F)=(−1)(p−1)(n+1)/2

K?
p∈N, p >2p
(p−1)! ≡ −1 mod p
H={a∈(Z/nZ)∗/a(n−1)/2≡a
n[n]}n
H= (Z/nZ)∗
n
k
1−2−k
q=prp r ∈N?I(n, q)
nFq
n∈N?Fq[X]Xqn−X=
Qd|nQP∈I(d,q)P
Fp
q=pnFp
nF4=F2[X]/(X2+X+ 1)
P∈Fq[X]n∈N?PFq
P Xqn−X r n P Xqn/r −X
Xp−X−a a ∈F?
pFp
X4+X+ 1 F2F4
X4+ 8X2+ 17X−1
Z
Fq

p q =ps
R∈Fq[X]SR:Q(X) mod R7−→ Q(Xq) mod R
FqFq[X]/(R)
qFq[X]/(R)
P∈Fq[X]
P
SP−Id (1, x, . . . , xdeg P−1)Fq[X]/(P)
x=Xmod P
r= dim(ker(SP−Id)) = deg(P)−(SP−Id)
P
r= 1
V∈Fq[X]P V mod P?∈ker(SP−
Id)
P=Qα∈Fq(P, V −α)
P∈Fq[X]Q P
α Q (P, P 0)α−1p-α α
p|α
P∈Fq[X] (P, P 0) = 1
(P, P 0) = P R ∈Fq[X]P=Rp
P0= 0
P∈Fq[X]
P
P P
(P, P 0)
(P, P 0) = 1
(P, P 0) = P R Rp=P
P(P, P 0)
P/ (P, P 0)
a k
2kn n >k
m a
m0m
Fqq=prk n 06k6n

n k Fq
C k Fn
q
n C
∀a= (a0, . . . , an−1)∈C, γ(a) = (an−1, a0, . . . , an−2)∈C
x, y ∈Fqd(x, y) =
w(x−y)w(x)x x
d C d = min{d(x, y)/?x, y ∈C, x 6=
y}Cbd−1
2c bd−1 2c
g=an−kXn−k+· · · +a0Xn−1
Fq[X]m= (a0, . . . , an−k,0,...,0) ∈Fn
qk
m, γ(m), . . . , γk−1(m)Fq
CFq
g
C