
theorem eq:
for F being Field holds F is maximal_algebraic iff F is algebraic-closed
proof
let F be Field;
A: now assume B: F is maximal_algebraic;
   now let p be irreducible Element of the carrier of Polynom-Ring F;
     consider E being F-finite FieldExtension of F such that
     C: deg(E,F) = deg p & p is_with_roots_in E by mi2;
     thus 1 = deg p by C,B,FIELD_7:8;
     end;
   hence F is algebraic-closed by al1;
   end;
now assume not F is maximal_algebraic;
  then consider E being F-algebraic FieldExtension of F such that
  B: not E == F;
  F is Subfield of E by FIELD_4:7; then
  C: the carrier of F c= the carrier of E by EC_PF_1:def 1;
  F: not deg(E,F) = 1 by B,FIELD_7:8;
  now assume G: for a being Element of E holds a in F;
    now let o be object;
      assume o in the carrier of E; then
      reconsider a = o as Element of E;
      a in F by G;
      hence o in the carrier of F;
      end;
    hence contradiction by F,FIELD_7:7,C,TARSKI:2;
    end; then
  consider a being Element of E such that D: not a in F;
  set p = MinPoly(a,F);
  E: deg(FAdj(F,{a}),F) = deg p by FIELD_6:67;
  a in {a} by TARSKI:def 1; then
  not {a} is Subset of F by D; then
  not deg(FAdj(F,{a}),F) = 1 by FIELD_7:8,FIELD_7:3;
  hence not F is algebraic-closed by E,al1;
  end;
hence thesis by A;
end;
