
theorem spl0:
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
ex E being FieldExtension of F
st FAdj(F,Roots(E,p)) is SplittingField of p
proof
let F be Field,
    p be non constant Element of the carrier of Polynom-Ring F;
consider E being FieldExtension of F such that
A: p splits_in E by FIELD_5:31;
take E;
set K = FAdj(F,Roots(E,p));
thus p splits_in K by A,lemma5;
now let U be FieldExtension of F;
  assume B: U is Subfield of K & p splits_in U;
  C: U is Subfield of E by B,EC_PF_1:5; then
  C1: E is FieldExtension of U by FIELD_4:7;
  D: F is Subfield of U by FIELD_4:7;
  E: Roots(U,p) c= Roots(E,p) by C1,lemma7; 
  Roots(E,p) c= Roots(U,p) by C1,A,B,lemma3; then
  Roots(E,p) is Subset of U by E,XBOOLE_0:def 10;
  hence K is Subfield of U by D,C,FIELD_6:37;
  end;
hence thesis;
end;
