
theorem R441:
for F being Field
for p being non zero Element of the carrier of Polynom-Ring F
holds p is reducible iff
      (p is Unit of Polynom-Ring F or
       ex q being monic Element of the carrier of Polynom-Ring F
                                 st q divides p & 1 <= deg q & deg q < deg p)
proof
let F be Field, p be non zero Element of the carrier of Polynom-Ring F;
now assume p is reducible; then
  per cases by RING_4:41;
  suppose p = 0_.(F);
    hence p is Unit of Polynom-Ring F or
          ex q being monic Element of the carrier of Polynom-Ring F
                                 st q divides p & 1 <= deg q & deg q < deg p;
    end;
  suppose p is Unit of Polynom-Ring F;
    hence p is Unit of Polynom-Ring F or
          ex q being monic Element of the carrier of Polynom-Ring F
                                 st q divides p & 1 <= deg q & deg q < deg p;
    end;
  suppose ex q being Element of the carrier of Polynom-Ring F
                         st q divides p & 1 <= deg q & deg q < deg p; then
    consider q being Element of the carrier of Polynom-Ring F such that
    A: q divides p & 1 <= deg q & deg q < deg p;
    set r = NormPolynomial q;
    q <> 0_.(F) by A,HURWITZ:20; then
    C: q is non zero by UPROOTS:def 5;
    deg r = deg q by REALALG3:11;
    hence p is Unit of Polynom-Ring F or
          ex q being monic Element of the carrier of Polynom-Ring F
                st q divides p & 1 <= deg q & deg q < deg p by A,RING_4:25,C;
    end;
  end;
hence thesis by RING_4:41;
end;
