
theorem lemNor32:
for F1 being Field,
    p1 being non constant Element of the carrier of Polynom-Ring F1
for F2 being FieldExtension of F1,
    p2 being non constant Element of the carrier of Polynom-Ring F2
for E being SplittingField of p2
for T being F1-algebraic Subset of F2 st T c= Roots(E,p2) & F2 == FAdj(F1,T)
holds p1 = p2 implies E is SplittingField of p1
proof
let F1 be Field,
    p1 be non constant Element of the carrier of Polynom-Ring F1;
let F2 be FieldExtension of F1,
    p2 be non constant Element of the carrier of Polynom-Ring F2;
let E be SplittingField of p2;
let T be F1-algebraic Subset of F2;
assume AS: T c= Roots(E,p2) & F2 == FAdj(F1,T);
reconsider E as F2-extending FieldExtension of F1;
reconsider T as finite F1-algebraic Subset of F2 by AS;
assume A: p2 = p1;
   p2 splits_in E by FIELD_8:def 1; then
consider a being non zero Element of E, q being Ppoly of E such that
C: p2 = a * q by FIELD_4:def 5;
D: p1 splits_in E by A,C,FIELD_4:def 5;
E: now let K be Field;
   assume p1 splits_in K; then
   consider a being non zero Element of K, q being Ppoly of K such that
   E: p1 = a * q by FIELD_4:def 5;
   thus p2 splits_in K by A,E,FIELD_4:def 5;
   end;
now let U being FieldExtension of F1;
  assume F1: p1 splits_in U & U is Subfield of E; then
  E is FieldExtension of U by FIELD_4:7; then
  F3: Roots(E,p1) c= the carrier of U by D,F1,FIELD_8:27;
  F1 is Subfield of U by FIELD_4:7; then
  F4: FAdj(F1,Roots(E,p1)) is Subfield of U by F1,F3,FIELD_6:37;
  reconsider T1 = T as Subset of E by AS,XBOOLE_1:1;
  T c= Roots(E,p1) by AS,A,FIELD_8:7; then
  F5: FAdj(F1,T1) is Subfield of FAdj(F1,Roots(E,p1)) by FIELD_7:10;
  FAdj(F1,T1) = FAdj(F1,T) by lemh1; then
  FAdj(F1,T) is Subfield of U by F4,F5,EC_PF_1:5; then
  U is FieldExtension of FAdj(F1,T) by FIELD_4:7; then
  F6: U is FieldExtension of F2 by AS,FIELD_12:37;
  p2 splits_in U by F1,E;
  hence E == U by F1,F6,FIELD_8:def 1;
  end;
hence thesis by A,C,FIELD_4:def 5,FIELD_8:def 1;
end;
