reserve n for Nat;

theorem
for F being Field
holds F is algebraic-closed iff
      (for p being non constant monic Polynomial of F holds p is Ppoly of F)
proof
let F be Field;
now assume AS: for p being non constant monic Polynomial of F
               holds p is Ppoly of F;
  now let p be Polynomial of F;
    assume A: len p > 1;
    then B: p is non zero by POLYNOM4:3;
    set np = NormPolynomial(p);
    (len np - 1) + 1 > 1 by A,POLYNOM5:57;
    then len np - 1 >= 1 by NAT_1:13;
    then np is non constant monic by B,HURWITZ:def 2;
    then reconsider np as Ppoly of F by AS;
    np is with_roots;
    hence p is with_roots by A,POLYNOM5:60;
    end;
  hence F is algebraic-closed by POLYNOM5:def 9;
  end;
hence thesis by cc3;
end;
