
theorem
for F being Field holds
F is algebraic-closed iff
for E being F-finite FieldExtension of F holds E == F
proof
let F be Field;
A: now assume F is algebraic-closed;
   then F is maximal_algebraic by eq;
   hence for E being F-finite FieldExtension of F holds E == F;
   end;
now assume not F is algebraic-closed; then
  not F is maximal_algebraic by eq;
  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);
  a in {a} by TARSKI:def 1; then
  not {a} is Subset of F by D; then
  not FAdj(F,{a}) == F by FIELD_7:3;
  hence ex E being F-finite FieldExtension of F st not E == F;
  end;
hence thesis by A;
end;
