Critère d`Eisenstein

Z[X]pgcd
1
Z[X]
A∈Z[X]A c(A)pgcd
A A B Z[X]
c(AB) = c(A)c(B)
A=anXn+. . . a1X+a0∈Z[X]p
(i)p a0
(ii)p a0, a1, . . . , an−1
(iii)p2a0
AQ[X]
A=
n
P
k=0
akXkB=
m
P
k=0
bkXkC=
m+n
P
k=0
ckXk=
AB A B C
p ckP∈Z[X]
P P (Z/pZ)[X]P=P
k∈N
skXkP=P
k∈N
skXksk
skp p ckC= 0 AB =AB =C= 0
(Z/pZ)[X]Z/pZA= 0 B= 0
p A B
Z[X]
AB =c(A)c(B)A
c(A)
B
c(B)
A
c(A)
B
c(B)
AB c(A)c(B)
AQ[X]A=BC B C
Z[X]A
α=c(A)A0=A/α ∈Z[X]A0A A0
A0=B0C0B0C0Q[X]
A β γ B0C0
B=βB0C=γC0Z[X]βγA0=BC
βγ =βγc(A0) = c(B)c(C)
A=α(B/β)(C/γ) = α(B/c(B))(C/c(C)) = (αB/c(B))(C/c(C))
αB/c(B)C/c(C)A
A B C Z[X]
n A =BC B =bkXk+. . . +b1X+b0C=
clXl+... +c1X+c0k= deg B l = deg C
A=BC Z/pZ[X]anXn=BC B C
k l bkcl=anp bk6= 0 cl6= 0
(Z/pZ)[X]B=bkXkC=clXlb0=c0= 0
p|b0p|c0p2a0=b0c0(iii)
Xn−2Q[X]n≥1p= 2
Q[X]