corps finis en master S1

ZnZ/nZ
GF (q)Fq
(Zn,+, .)n
⇒(Zn,+, .)n=ab 1< a, b <
n b =a−1ab =a−1n= 0 mod n
⇐n1≤m≤n−1
n u v
un+vm = 1 Zn¯v¯m≡¯ 1 ¯m(Zn,+, .)
KF
FK F [K:F]
F F
Q F Zp
p
f:Z−→ F
f(1) = 1 fFf(Z)
Z Z Q f
p f(p) = 0 ker(f) = pZ
f(Z)Zpp p =rs
1< r, s < p f(p) = f(rs) = f(r)f(s) = 0 F
F F
R R
m > 0ma = 1 + · · · + 1 = 0
m m ∈N∗m.16= 0 R
Fp(a+b)pi=
api+bpia, b ∈Fi∈N∗

p
p
kp(p−1) · · · (p−k+ 1) = k!p kp
k! 0 < k < p p p
k
i
F
F
F F prp
r > 0F
Zp
F F
ZpFpF
Zpr ω1,· · · , ωrF Zp
Fa1.ω1+· · · +
ar.ωrai∈Zpq=pr
FqZpFq
FpZp
F
ZpFp
Zp
p r > 0
pr
F F[x]
IF[x]I= (p(x)) p(x)
I
Fp(x)∈F[x]p(x)
((p(x)) = (p(x))F[x] = {p(x)g(x)|g(x)∈F[x]}
F[x]/(p(x) = {f(x) + I|f(x)∈F[x]}
f(x) + I f(x)p(x) = 0
f(x) + g(x) = f(x) + g(x)
f(x).g(x) = f(x).g(x)
KFp(x)∈F[x]
FK=F[x]/(p(x))
KF

f(x)∈F[x]f(x)6= 0 p(x)
p(x)f(x)pgcd(f(x), p(x)) = 1 ∈
F∗a(x)b(x)F[x]a(x)f(x) + b(x)p(x) = 1
¯a(x)¯ f(x) + ¯ b(x)¯p(x) = ¯ 1p(x) = 0 K
K
F⊂F[x]F
φ:F→K
a→a
Im(φ) = FK
F
KFK
F¯a a ∈F
p(x)∈F[x]m≥1p(x) =
c0+c1x+· · · +cmxmf(x)∈F[x]q(x)r(x)
F(x)f(x) = q(x)p(x) + r(x)deg(r)< m r(x)
r(x) = a0+a1x+· · · +am−1xm−1ai∈F
p(x) = ¯ 0
f(x) = q(x).p(x) + r(x) = a0+a1x+· · · +am−1.xm−1
t=x f(x) = a0+a1t+· · · +am−1.tm−1
Fp(x) = c0+c1x+· · ·+cmxm∈F[x]
m≥1K=F[x]/(p(x))
K=a0+a1t+· · · +am−1tm−1/a0,· · · , am−1∈F;p(t)=0
a(t) = a0+· · · +am−1tm−1b(t) = b0+· · · +bm−1t−1
a(t)b(t) = a0b0+ (a0b1+a1b0) + · · · +am−1bm−1t2m−2
a(t)b(t) = q(t)(c0+c1t+· · · +cmtm) + d0+d1t+· · · +dm−1tm−1
a(t)b(t) = d0+d1t+· · · +dm−1tm−1
K=R[x]/(x2+ 1)
Cp(x) = x2+ 1
RK K
K={a0+a1t/a0, a1∈R}t t2+ 1 = 0
f:K→Cf(a+bt) = a+ib
prZp
p(x)∈Zp[x]r p(x) = c0+c1x+
· · · +xrK=Zp[x]/p(x)K=
a0+a1t+· · · +ar−1tr−1|ai∈Zp, p(t)=0K pr
Fn n