
theorem
for F being Field holds
F is algebraic-closed iff
for p being non constant Polynomial of F holds p splits_in F
proof
let F be Field;
A: now assume B: F is algebraic-closed;
   now let p be non constant Polynomial of F;
     reconsider p1 = p as Element of the carrier of Polynom-Ring F
       by POLYNOM3:def 10;
     deg p1 > 0 by RATFUNC1:def 2; then
     reconsider p1 = p as non constant
                                  Element of the carrier of Polynom-Ring F
       by RING_4:def 4;
     set q = NormPolynomial p1;
     deg q = deg p by RING_4:27,RING_4:29; then
     reconsider q as non constant monic Polynomial of F
                                   by RING_4:def 4,RATFUNC1:def 2;
     reconsider q as Ppoly of F by B,RING_5:70;
     C: LC p <> 0.F;
     q = (LC p)" * p by RING_4:23; then
     LC p * q = ((LC p) * (LC p)") * p by RING_4:11
             .= ((LC p)" * (LC p)) * p by GROUP_1:def 12
             .= (1.F) * p  by C,VECTSP_1:def 10
             .= p;
     hence p splits_in F by FIELD_4:def 5;
     end;
   hence for p being non constant Polynomial of F holds p splits_in F;
   end;
now assume B: for p being non constant Polynomial of F holds p splits_in F;
  now let p be non constant monic Polynomial of F;
    consider a being non zero Element of F, q being Ppoly of F such that
    C: p = a * q by B,FIELD_4:def 5;
    1.F = LC p by RATFUNC1:def 7
       .= a * LC q by C,RING_5:5
       .= a * 1.F by RING_5:50;
    hence p is Ppoly of F by C;
    end;
  hence F is algebraic-closed by RING_5:70;
  end;
hence thesis by A;
end;
