
theorem lemppolspl3:
for F being Field
for p being non constant Polynomial of F
for q being non zero Polynomial of F
st p *' q splits_in F holds p splits_in F
proof
let F be Field;
let p be non constant Polynomial of F;
let q be non zero Polynomial of F;
assume p *' q splits_in F; then
consider a being non zero Element of F, u being Ppoly of F such that
A: p *' q = a * u by FIELD_4:def 5;
set b = (LC p) * a";
B: LC p <> 0.F & a" <> 0.F by  VECTSP_2:13;
C2: p is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
not deg p <= 0 by RATFUNC1:def 2; then
C1: len p - 1 > 0 by HURWITZ:def 2; then
len p <> 0; then
len(NormPolynomial p) = len p by POLYNOM5:57; then
deg(NormPolynomial p) > 0 by C1,HURWITZ:def 2; then
NormPolynomial p is non constant monic by RATFUNC1:def 2; then
C: (LC p)" * p is non constant monic by C2,RING_4:23;
D: b is non zero by B,VECTSP_2:def 1;
E: (LC p)" is non zero by VECTSP_2:13; then
F: (LC p)" * (LC p) = 1.F by VECTSP_1:def 10;
a" <> 0.F by  VECTSP_2:13; then
a" * a = 1.F by VECTSP_1:def 10; then
u = (a" * a) * u
 .= a" * (p *' q) by A,RING_4:11
 .= ((LC p)" * (LC p)) * (p *' (a" * q)) by F,RING_4:10
 .= (LC p)" * ((LC p) * (p *' (a" * q))) by RING_4:11
 .= ((LC p)") * (p *' ((LC p) * (a" * q))) by RING_4:10
 .= ((LC p)") * (p *' (b * q)) by RING_4:11
 .= ((LC p)" * p) *' (b * q) by RING_4:10; then
(LC p)" * p is Ppoly of F by D,C,lemppolspl3a;
hence thesis by E,lemppolspl2,lemppolspl;
end;
