
theorem alg2:
for F being Field holds
F is algebraic-closed iff
for p being non constant Polynomial of F holds p is with_roots
proof
let F be Field;
now assume B: for p being non constant Polynomial of F holds p is with_roots;
  now let p be Polynomial of F;
    assume C: len p > 1;
    deg p = len p - 1 by HURWITZ:def 2;
    then deg p + 1 - 1 > 1 - 1 by C,XREAL_1:9;
    then p is non constant by RATFUNC1:def 2;
    hence p is with_roots by B;
    end;
  hence F is algebraic-closed by POLYNOM5:def 9;
  end;
hence thesis;
end;
