
theorem lemma5:
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
for E being FieldExtension of F 
st p splits_in E holds p splits_in FAdj(F,Roots(E,p))
proof
let F be Field,
    p be non constant Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F;
assume A: p splits_in E;
set K = FAdj(F,Roots(E,p));
B: E is K-extending by FIELD_4:7;
Roots(E,p) is Subset of K by FIELD_6:35; 
hence thesis by A,B,lemma6;
end;
