
theorem ZZ3x:
for F being Field
for p being monic Polynomial of F st p divides 1_.F holds p = 1_.F
proof
let F be Field, r be monic Polynomial of F;
assume AS: r divides 1_.F;
reconsider r1 = r as Element of the carrier of Polynom-Ring F
   by POLYNOM3:def 10;
deg r1 <= deg(1_.F) by AS,RING_5:13; then
deg r1 <= 0 by RATFUNC1:def 2; then
consider a being Element of F such that
A: r1 = a|F by RING_4:def 4,RING_4:20;
1.F = LC r by RATFUNC1:def 7 .= a by A,RING_5:6;
hence thesis by A,RING_4:14;
end;
