
theorem
for F being Field holds
F is algebraic-closed iff
for p,q being Element of the carrier of Polynom-Ring F
holds p,q are_coprime iff p,q have_no_common_roots
proof
let F be Field;
A: now assume Z: F is algebraic-closed;
   now let p,q be Element of the carrier of Polynom-Ring F;
      set g = p gcd q;
      assume AS: not p,q are_coprime;
      per cases;
      suppose X: p = 0_.(F) & q = 0_.(F);
        eval(0_.(F),0.F) = 0.F by POLYNOM4:17; then
        0.F is_a_root_of p & 0.F is_a_root_of q by X,POLYNOM5:def 7;
        hence p,q have_a_common_root by RATFUNC1:def 4,RATFUNC1:def 3;
        end;
      suppose K: p <> 0_.(F) or q <> 0_.(F);
        now assume K0: g is constant;
          per cases;
          suppose K1: g = 0_.(F);
            consider r1 being Polynomial of F such that
            K2: (0_.(F)) *' r1 = p by K1,RING_4:52,RING_4:1;
            consider r2 being Polynomial of F such that
            K3: (0_.(F)) *' r2 = q by K1,RING_4:52,RING_4:1;
            thus contradiction by K,K2,K3;
            end;
          suppose K1: g <> 0_.(F);
            reconsider p1 = p ,q1 = q as Polynomial of F;
            reconsider g1 = g as Element of the carrier of Polynom-Ring F
               by POLYNOM3:def 10;
            deg g <= 0 by K0,RATFUNC1:def 2; then
            reconsider g1 as
                  non zero constant Element of the carrier of Polynom-Ring F
               by K1,UPROOTS:def 5,RING_4:def 4;
            consider a being Element of F such that
            K3: g1 = a|F by RING_4:20;
            per cases by K;
            suppose p <> 0_.(F); then
              p1 is non zero by UPROOTS:def 5; then
              1.F = LC(a|F) by K3,RATFUNC1:def 7 .= a by RING_5:6;
              then g = 1_.(F) by K3,RING_4:14;
              hence contradiction by AS,copr1;
              end;
            suppose q <> 0_.(F); then
              q1 is non zero by UPROOTS:def 5; then
              1.F = LC(a|F) by K3,RATFUNC1:def 7 .= a by RING_5:6;
              then g = 1_.(F) by K3,RING_4:14;
              hence contradiction by AS,copr1;
              end;
            end;
          end; then
        consider a being Element of F such that
        F: a is_a_root_of g by Z,POLYNOM5:def 8;
        X: eval(g,a) = 0.F by F,POLYNOM5:def 7;
        consider gp being Polynomial of F such that
        G: g *' gp = p by RING_4:52,RING_4:1;
        consider gq being Polynomial of F such that
        J: g *' gq = q by RING_4:52,RING_4:1;
        eval(p,a) = 0.F * eval(gp,a) &
        eval(q,a) = 0.F * eval(gq,a) by J,G,X,POLYNOM4:24; then
        a is_a_root_of p & a is_a_root_of q by POLYNOM5:def 7;
        hence p,q have_a_common_root by RATFUNC1:def 4,RATFUNC1:def 3;
        end;
     end;
   hence for p,q being Element of the carrier of Polynom-Ring F
      holds p,q are_coprime iff p,q have_no_common_roots by copr2;
   end;
now assume not F is algebraic-closed; then
  reconsider F1 = F as non algebraic-closed Field;
  set p = the non constant non with_roots monic
                                Element of the carrier of Polynom-Ring F1;
  reconsider p1 = p as Element of Polynom-Ring F1;
  reconsider e = 1.(Polynom-Ring F1) as Element of Polynom-Ring F1;
  H: e = 1_.(F1) by POLYNOM3:def 10;
  B: not p,p are_coprime
     proof
     D: p1 divides p1;
     now assume 1.(Polynom-Ring F1) is a_gcd of p1,p1; then
       p1 divides e by D,RING_4:def 10; then
       p divides 1_.(F1) by H; then
       deg p <= deg 1_.(F) by RING_5:13; then
       deg p <= 0 by RATFUNC1:def 2;
       hence contradiction by RING_4:def 4;
       end;
      hence thesis;
     end;
  now assume p,p have_a_common_root; then
    consider a being Element of F1 such that
    A: a is_a_common_root_of p,p by RATFUNC1:def 4;
    a is_a_root_of p & a is_a_root_of p by A,RATFUNC1:def 3;
    hence contradiction by POLYNOM5:def 8;
    end;
  hence ex p,q being Element of the carrier of Polynom-Ring F
     st (not p,q are_coprime) & p,q have_no_common_roots by B;
  end;
hence thesis by A;
end;
