
theorem ft1:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
holds E == FAdj(F,{a}) iff deg MinPoly(a,F) = deg(E,F)
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E;
A: now assume A0: E == FAdj(F,{a});
   deg(FAdj(F,{a}),F) = deg MinPoly(a,F) by FIELD_6:67;
   hence deg MinPoly(a,F) = deg(E,F) by A0,str11;
   end;
now assume B0: deg MinPoly(a,F) = deg(E,F);
   reconsider Fa = FAdj(F,{a}) as FieldExtension of F;
   reconsider E1 = E as Fa-extending FieldExtension of F by FIELD_4:7;
   VecSp(E1,F) is finite-dimensional by B0,FIELD_4:def 7; then
   E1 is F-finite by FIELD_4:def 8; then
   reconsider E1 = E as
      F-extending FAdj(F,{a})-finite FieldExtension of FAdj(F,{a}) by alg0;
   reconsider d1 = deg(E,F) as non zero Nat by B0;
   deg(E,F) = deg(E1,FAdj(F,{a})) * deg(FAdj(F,{a}),F) by degmult
           .= deg(E1,FAdj(F,{a})) * deg(E,F) by B0,FIELD_6:67;
  :: then deg(E1,FAdj(F,{a})) = 1 by XCMPLX_1:7;
   hence E == FAdj(F,{a}) by str1a,XCMPLX_1:7;
   end;
hence thesis by A;
end;
