Factorisation dans Fp[x] 1 Le cas sans facteurs multiples

Fp[x]
Fp=Z/pZp
Z[x]
Fp[x]f∈Fp[x]
f=cfm1
1· · · fms
s,
c∈F∗
pmi∈N∗fi∈Fp[x]
f∈Fp[x]
[c, (f1, m1),...,(fs, ms)]
mi= 1
•
•
f mi= 1 i

f∈Fp[x]
R=Fp[x]/(f).
Fpd:= deg(f)fi
Ri=Fp[x]/(fi)
Fpdeg(fi)fi
r:R−→ R1× · · · × Rs
gmod f7−→ (gmod f1, . . . , g mod fs)
rir= (r1, . . . , rs)
φp
φp:R→R
g7→ gp
FpR p
(f+g)p=fp+gp,
pj
pCj
p1≤j≤p−1
p
B ⊂ R φp
B:= ker(φp−Id),
Id R r = (r1, . . . , rs)
g∈ B ⇐⇒ ri(g)p=ri(g)∀i.
KFpFp
KFp={z∈K, zp=z}
z∈Fpz= 0 zp=z z ∈Fp\ {0}
F∗
pFpFp
p−1z p −1zp−1= 1 zp=z K

zp−z= 0 p p p
FpFp
g∈ B ri(g)∈Fp, i = 1, . . . , s r
r:B −→ Fs
p
BFps
f s
φp−Id (1, x, . . . , xn−1)R
s > 1
Fp[x]
xp−x=Y
ω∈Fp
(x−ω)
g∈Fp[x] deg(g)≥1
gp−g=Y
ω∈Fp
(g−ω).
g−ω
pgcd(f, gp−g) = Y
ω∈Fp
pgcd(f, g −ω).
¯g:= gmod (f)Bpgcd(f, gp−g) = f
f
f=Y
ω∈Fp
pgcd(f, g −ω).
¯g∈R\Fpg
(f)
¯g∈ B ω∈Fppgcd(f, g −ω)
f1f
ωpgcd(f, g −ω)f g
(f) pgcd(f, g −ω)6=fpgcd(f, g −ω)
f
s > 1fpgcd(f, g −ω)
gBω∈Fpf=f1f2
f1f2

pgcd(f, g −ω)gB
ωFpp
p p
f g −ω gp−g
gp−g=g(gp−1
2−1)(gp−1 2+ 1)
p6= 2 ¯g∈ B f gp−g
f= pgcd(f, gp−g) = pgcd(f, g) pgcd(f, g p−1
2−1) pgcd(f, g p−1 2+ 1).
¯g6= 0 g1:= pgcd(f, g)6= 1 g1f
g1f/g1g1= 1 g2:= pgcd(f, g p−1
2−1) = f
gp−1
2−1≡0 mod (fi)∀i= 1, . . . , s
g2= 1 g3:= pgcd(f, g p−1
2+ 1) = f
gp−1
2+ 1 ≡0 mod (fi)∀i= 1, . . . , s.
g2∈ {1, f}
p6= 2
S+:= {z∈F∗
p, z p−1
2−1=0}et S−:= {z∈F∗
p, z p−1
2+ 1 = 0}
(p−1)/2F∗
p
0 = zp−1−1 = (zp−1
2−1)(zp−1 2+ 1) ∀z∈F∗
p.
pF∗
p=S+∪S−
Card(S+) + Card(S−) = Card(F∗
p) = p−1.
x(p−1)/2−1x(p−1)/2+1 Fp[x] (p−1)/2
Card(S+)≤p−1
2et Card(S−)≤p−1 2.
¯g∈ B gmod (fi)∈Fp⊂Fp[x]/(fi)i∈F∗
p
¯g∈ B B Fp
gmod (fi)∈FpFp1/p i = 1, . . . , s
gmod (fi)6= 0 g(p−1)/2−1 = 0 mod fi
g(p−1)/2+ 1 = 0 mod fi1/2

1/2sg2f1/2(s−1)
1/2s≥2g2
g k
1/2k(s−1) ≤1/2k,
f1k
1 2
Fpf∈Fp[x]
1f f Fp[x]
1
1xp−x
1
xp−x=Y
ω∈Fp
(x−ω).
h:= pgcd(xp−x, f)
1f
h1f
f∈Fp[x]
f∈Fp[x]d d
Fp
K:= Fp[x]/(f).