
theorem
for F being Field
for p being Element of the carrier of Polynom-Ring F
holds p gcd 0_.(F) = NormPolynomial p
proof
let F be Field, p be Element of the carrier of Polynom-Ring F;
set s = NormPolynomial p;
per cases;
suppose A0: p is zero; then
   A1: p = 0_.(F) by UPROOTS:def 5;
   thus p gcd 0_.(F) = 0_.(F) by A0,UPROOTS:def 5
     .= s by A1,RING_4:22;
   end;
suppose S: p is non zero;
   p *' 1_.(F) = p; then
B: s divides p by RING_4:25,RING_4:1;
   0_.(F) = s *' 0_.(F); then
C: s divides 0_.(F) by RING_4:1;
now let r be Polynomial of F;
   assume D: r divides 0_.(F) & r divides p;
   r is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
   hence r divides s by D,RING_4:26;
   end;
hence thesis by S,C,B,RING_4:53;
end;
end;
