
theorem ZZ3:
for F being Field
for p being Ppoly of F
for q being Polynomial of F
holds (Roots p) /\ (Roots q) = {} iff p gcd q = 1_.F
proof
let F be Field;
let p be Ppoly of F; let q be Polynomial of F;
Z: now assume AS: (Roots p) /\ (Roots q) = {};
   (1_.F) *' p = p & (1_.F) *' q = q; then
A: 1_.F divides p & 1_.F divides q by RING_4:1;
now let r be Polynomial of F;
  assume C: r divides q & r divides p; then
  consider u being Polynomial of F such that
  D: p = r *' u by RING_4:1;
  Roots r c= Roots q & Roots r c= Roots p by C,ZZ3b; then
  E: Roots r c= Roots q /\ Roots p;
  now assume F: r is non constant; then
    F2: r is non zero Element of the carrier of Polynom-Ring F
        by POLYNOM3:def 10;
    set r1 = NormPolynomial r;
    set u1 = (LC r) * u;
    F3: LC r <> 0.F by F;
    F1: r1 *' u1
           = ((LC r)" * r) *' ((LC r) * u) by F2,RING_4:23
          .= (LC r) * (((LC r)" * r) *' u) by RING_4:10
          .= (LC r) * ((LC r)" * (u *' r)) by RING_4:10
          .= ((LC r) * (LC r)") * (u *' r) by RING_4:11
          .= 1.F * (r *' u) by F3,VECTSP_1:def 10
          .= p by D;
    F4: deg r > 0 by F,RATFUNC1:def 2;
    F5: deg r = len r - 1 by HURWITZ:def 2; then
    I: len r <> 0 by F; then
    len r1 = len r by POLYNOM5:57; then
    deg r1 = deg r by F5,HURWITZ:def 2; then
    K: r1 is non constant monic by F,F4,RATFUNC1:def 2;
    u1 is non zero by F1; then
    G: r1 is Ppoly of F by F1,K,FIELD_8:10;
    Roots r1 = Roots r by I,POLYNOM5:61;
    hence contradiction by AS,E,G;
    end; then
  deg r <= 0 by RATFUNC1:def 2; then
  r is constant Element of the carrier of Polynom-Ring F
     by POLYNOM3:def 10,RING_4:def 4; then
  consider a being Element of F such that
  F6: r = a|F by RING_4:20;
  F7: now assume a = 0.F;
      then r = 0_.(F) by F6,RING_4:13;
      hence contradiction by E,AS;
      end;
  (a|F) *' ((a")|F)
     = (a * (a"))|F by RING_4:18
    .= (1.F)|F by F7,VECTSP_1:def 10
    .= 1_.F by RING_4:14;
  hence r divides 1_.F by F6,RING_4:1;
  end;
hence p gcd q = 1_.F by A,RING_4:53;
end;
now assume AS: (Roots p) /\ (Roots q) <> {};
  set a = the Element of (Roots p) /\ (Roots q);
  A1: a in (Roots p) & a in (Roots q) by AS,XBOOLE_0:def 4; then
  reconsider a as Element of F;
  a is_a_root_of p by A1,POLYNOM5:def 10;then
  A2: eval(p,a) = 0.F by POLYNOM5:def 7;
  a is_a_root_of q by A1,POLYNOM5:def 10; then
  eval(q,a) = 0.F by POLYNOM5:def 7; then
  A3: rpoly(1,a) divides p & rpoly(1,a) divides q by A2,RING_5:11;
  deg rpoly(1,a) = 1 & deg(1_.F) <= 0 by HURWITZ:27,RATFUNC1:def 2;
  hence p gcd q <> 1_.F by A3,RING_5:13,RING_4:52;
  end;
hence thesis by Z;
end;
