
theorem
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
for E being FieldExtension of F
for U being FieldExtension of E holds p splits_in E implies p splits_in U
proof
let F be Field,
    p be non constant Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F;
let U be FieldExtension of E;
assume p splits_in E; then
consider a being non zero Element of E, q being Ppoly of E such that
A: p = a * q by FIELD_4:def 5;
B: E is Subfield of U by FIELD_4:7; then
the carrier of E c= the carrier of U by EC_PF_1:def 1; then
reconsider b = a as Element of U;
a <> 0.E & 0.U = 0.E by B,EC_PF_1:def 1; then
reconsider b as non zero Element of U by STRUCT_0:def 12;
reconsider r = q as Ppoly of U by lemmapp;
C: b|U = a|E by FIELD_6:23;
b * r = (b|U) *' r by poly1
     .= (a|E) *' q by C,FIELD_4:17
     .= a * q by poly1;
hence thesis by A,FIELD_4:def 5;
end;
