
theorem simpAgcd2:
for F being Field
for a being Element of F, p being Ppoly of F,{a} holds p = rpoly(1,a)
proof
let F be Field, a be Element of F, p be Ppoly of F,{a};
deg p = card {a} by RING_5:60 .= 1 by CARD_2:42; then
consider x,z being Element of F such that
A: x <> 0.F & p = x * rpoly(1,z) by HURWITZ:28;
B: 1.F = LC p by RATFUNC1:def 7
      .= x * LC rpoly(1,z) by A,RING_5:5
      .= x * 1.F by RING_5:9;
{a} = Roots p & Roots rpoly(1,z) = {z} by RING_5:18,RING_5:63; then
{a} = {z} & a in {a} by A,B,TARSKI:def 1;
hence thesis by A,B,TARSKI:def 1;
end;
