
theorem thirred:
for F being Field
for p being Element of the carrier of Polynom-Ring F st deg p = 3
holds p is reducible iff p is with_roots
proof
let F be Field, p be Element of the carrier of Polynom-Ring F;
assume A: deg p = 3; then
reconsider p as
  non linear non constant Element of the carrier of Polynom-Ring F
  by FIELD_5:def 1,RING_4:def 4;
now assume B0: p is reducible;
  p <> 0_.(F) & not p is Unit of Polynom-Ring F; then
  consider q being Element of the carrier of Polynom-Ring F such that
  B1: q divides p & 1 <= deg q & deg q < deg p by B0,RING_4:41;
  consider r being Polynomial of F such that
  B2: p = q *' r by B1,RING_4:1;
  B3: deg q is Element of NAT by B1,INT_1:3;
  q <> 0_.(F) & r <> 0_.(F) by B2; then
  B4: deg p = deg q + deg r by B2,HURWITZ:23;
  B5: deg q < 2 + 1 by B1,A;
  deg q <= 2 implies deg q = 0 or ... or deg q = 2 by B3; then
  per cases by B5,B3,NAT_1:13;
  suppose deg q = 0;
    hence p is with_roots by B1;
    end;
  suppose deg q = 1; then
    q is linear by FIELD_5:def 1;
    hence p is with_roots by B2;
    end;
  suppose deg q = 2; then
    r is linear by B4,A,FIELD_5:def 1;
    hence p is with_roots by B2;
    end;
  end;
hence thesis;
end;
