
theorem F811:
for F being Field,
    E being FieldExtension of F
for p being non constant Polynomial of F
for q being non zero Polynomial of F
st p *' q splits_in E holds p splits_in E
proof
let F be Field, E be FieldExtension of F;
let p be non constant Polynomial of F;
let q be non zero Polynomial of F;
assume p *' q splits_in E; then
consider b being non zero Element of E, s being Ppoly of E such that
A: p *' q = b * s by FIELD_4:def 5;
B: F is Subring of E by FIELD_4:def 1; then
reconsider p1 = p, q1 = q as Polynomial of E by FIELD_4:9;
p is Element of the carrier of Polynom-Ring F &
p1 is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10; then
deg p > 0 & deg p = deg p1 by RATFUNC1:def 2,FIELD_4:20; then
reconsider p1 as non constant Polynomial of E by RATFUNC1:def 2;
q <> 0_.(F) & 0_.(E) = 0_.(F) by B,C0SP1:def 3;
then reconsider q1 as non zero Polynomial of E by UPROOTS:def 5;
p *' q = p1 *' q1 by FIELD_4:17;
then p1 splits_in E by A,FIELD_4:def 5,FIELD_8:11;
then consider a being non zero Element of E, r being Ppoly of E such that
C: p1 = a * r by FIELD_4:def 5;
thus thesis by C,FIELD_4:def 5;
end;
