112: Corps finis. Applications.

Z/pZ
Z/nZ
K∀x∈K, xq=x
K K∗Z/(q−1)Z
K∗
Z/pZ⊂K
K p
Z/pZ|K|=pn
n= [K:Z/pZ]
q=pnq
Xq−XFp
Fq
FpnZ/pZ
F4'Z/2Z[X]/(1 + X+X2)F8'Z/2Z[X]/(1 + X+X3)F9'Z/3Z[X]/(X2+ 1)
F16'Z/2Z[X]/(1 + X+X4)

nFpFpkk n
Kd
pXq−X=
Q
d|n
(Q
Q∈Kd
pQ
Q)
Kd
ppn=P
d|n
dI(p, d)
q n Φn
FqZ/nZ∗
K p x 7→ xp
K
p
K
x7→ x2K∗−1,1q+ 1/2
xq−1
2= 1 x
xq−1
2=−1
−1K
p= 4m+ 1
L K [L:K
L K −ev
K[X]
Fpn⊂Fpmn|m
F8F64

FpnFpdd|n
FpnFpn
Fpdn/d d −ime
FpnFpdFpn
Fpn
Z/pZ
Fp
pFp
x2−a= 0 Fp
x2−5y2x y
(p
q) = pq−1
2
Z/pZ{−1,1}
(2
p)=(−1)p2
−1
8
(p
q)( q p)=(−1)(p−1 2
q−1
2)
F17F1987 Fp
p20k+ 1,3,7,9
Z Fp
Fp
ax2+by2=c
p P ∈Z[X]P=anXn+
. . . +a0¯ P=Cp(an)Xn+. . . +Cp(a0)¯ PZ/pZ[X]
Q[X]Z[X]
X3+ 462X2+ 2433X−67691 Z Z/2Z[X]
p Xp−X−1Z Z/pZ
X4+ 1 FpFp
X5+X2+X+ 2 ZX5−7Z

Fpp
Q Z
ΦnFp
Z/nZ∗
n= 2,4, qα,2qα q
E dim(E) = qn
pn(n+1)/2
pn(n−1)/2
q
SnGLn(k)σ
δi, σ(i)
Gln(Fp)
Fpε(u) = det(u)/p
|GLn(Fqn)|= (qn−1) . . . (qn−qn−1),
|SLn(Fqn)|= (qn−1) . . . (qn−qn−2)qn−1,
|P GLn(Fqn)|= (qn−1) . . . (qn−qn−2)qn−1,
|P SLn(Fqn)|= (qn−1) . . . (qn−qn−2)qn−1/pgcd(n, q −1).
GL2(F2)'P GL2(F2)'SL2(F2)'P SL2(F2)' S3
P GL2(F3)' S4
P SL2(F3)' A4
P GL2(F4)'P SL2(F4)' A5
P SL2(F5)' A5
P SL2(Fq)q>4
P SL2(F7)
P SL3(F2)

E n Q
{0}
1 0 . . . 0
. . .
0. . . α
k
σ f :E×E→k
x7→ f(x, y)
y7→ f(x, y)f(λx, y) = σ(λ)f(x, y)σ
f(y, x) = σ(f(x, y))
k k
σ k =F2
pk=F2 p
σ(x) = xqFq
σ