
theorem alg1:
for F being Field,
    E being FieldExtension of F
for a being Element of E
holds a is F-algebraic iff
      ex B being F-finite FieldExtension of F st E is B-extending & a in B
proof
let F be Field, E be FieldExtension of F; let a be Element of E;
A: now assume A0: a is F-algebraic;
   A1: {a} is Subset of FAdj(F,{a}) by FIELD_6:35;
   FAdj(F,{a}) is Subring of E by FIELD_5:12; then
   A2: E is FAdj(F,{a})-extending by FIELD_4:def 1;
   a in {a} by TARSKI:def 1;
   then a in FAdj(F,{a}) by A1;
   hence ex B being F-finite FieldExtension of F
         st E is B-extending & a in B by A0,A2;
   end;
now assume ex B being F-finite FieldExtension of F
           st E is B-extending & a in B; then
  consider B being F-finite FieldExtension of F such that
  A0: E is B-extending & a in B;
  reconsider E1 = E as B-extending FieldExtension of F by A0;
  reconsider a1 = a as B-membered Element of E1 by A0,defmem;
  @(@(B,a1),E1) is F-algebraic;
  hence a is F-algebraic;
  end;
hence thesis by A;
end;
