
theorem divfin2h:
for F being Field
for a being Element of F holds MonicDivisors rpoly(1,a) = { 1_.F, rpoly(1,a) }
proof
let F be Field, a be Element of F;
set M = {1_.F,rpoly(1,a)};
H: (1_.F) is Element of the carrier of Polynom-Ring F &
   rpoly(1,a) is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
A: now let o be object;
   assume o in M; then
   per cases by TARSKI: def 2;
   suppose B: o = 1_.(F);
     (1_.(F)) *' rpoly(1,a) = rpoly(1,a); then
     1_.F divides rpoly(1,a) by RING_4:1;
     hence o in MonicDivisors rpoly(1,a) by B,H;
     end;
   suppose B: o = rpoly(1,a);
     (1_.(F)) *' rpoly(1,a) = rpoly(1,a); then
     rpoly(1,a) divides rpoly(1,a) by RING_4:1;
     hence o in MonicDivisors rpoly(1,a) by B,H;
     end;
   end;
now let o be object;
  assume o in MonicDivisors rpoly(1,a); then
  consider q being monic Element of the carrier of Polynom-Ring F such that
  B1: o = q & q divides rpoly(1,a);
  deg q <= deg rpoly(1,a) by B1,RING_5:13; then
  deg q <= 1 by HURWITZ:27; then
  per cases by XXREAL_0:1;
  suppose deg q = 1; then
    consider x,b being Element of F such that
    B2: x <> 0.F & q = x * rpoly(1,b) by HURWITZ:28;
    B3: 1.F = LC q by RATFUNC1:def 7
           .= x * (LC rpoly(1,b)) by B2,RING_5:5
           .= x * 1.F by RATFUNC1:def 7; then
    a = b by B1,B2,RING_5:21;
    hence o in M by B1,B2,B3,TARSKI:def 2;
    end;
  suppose deg q < 1; then
    deg q + 1 <= 1 by INT_1:7; then
    deg q + 1 - 1 <= 1 - 1 by XREAL_1:9; then
    consider b being Element of F such that B2: q = b|F
by RING_4:def 4,RING_4:20;
    1.F = LC q by RATFUNC1:def 7 .= b by B2,RING_5:6; then
    q = 1_.F by B2,RING_4:14;
    hence o in M by B1,TARSKI:def 2;
    end;
  end;
hence thesis by A,TARSKI:2;
end;
