
theorem T88b:
for F being Field
for p being Element of the carrier of Polynom-Ring F
holds p is reducible iff
      (p = 0_.(F) or p is Unit of Polynom-Ring F or
       ex q being Element of the carrier of Polynom-Ring F
                                 st q divides p & 1 <= deg q & deg q < deg p)
proof
let R be Field, p be Element of the carrier of Polynom-Ring R;
set K = Polynom-Ring R;
reconsider pp = p as Polynomial of R;
A: now assume p is reducible;
   then AS: p = 0.K or p is Unit of K or
      ex a being Element of K st a divides p &
        not(a is Unit of K) & not(a is_associated_to p) by RING_2:def 9;
   thus p = 0_.(R) or p is Unit of Polynom-Ring R or
        ex q being Element of the carrier of Polynom-Ring R
                                 st q divides p & 1 <= deg q & deg q < deg p
     proof
     assume A0: not(p = 0_.(R)) & not(p is Unit of K);
     then consider a being Element of K such that
     A1: a divides p & not(a is Unit of K) & not(a is_associated_to p)
         by AS,POLYNOM3:def 10;
     reconsider q = a as Polynomial of R by POLYNOM3:def 10;
     q divides pp by A1; then
     consider r being Polynomial of R such that
     A2: pp = q *' r by T2;
     reconsider rr = r as Element of K by POLYNOM3:def 10;
     A10: p = a * rr by A2,POLYNOM3:def 10;
     A5: q <> 0_.(R) by A0,A2,POLYNOM3:34;
     A6: r <> 0_.(R) by A0,A2,POLYNOM3:34;
     then A9: deg p = (deg q) + (deg r) by A2,A5,HURWITZ:23;
     A11: deg q is Element of NAT & deg r is Element of NAT by A5,A6,T8b;
     then A7: deg q <= deg p by A9,NAT_1:11;
     A3: now assume deg q = deg p;
         then rr is Unit of K by A9,T8;
         hence contradiction by A1,A10,GCD_1:18;
         end;
     thus ex b being Element of the carrier of K
                 st b divides p & 1 <= deg b & deg b < deg p
       proof
       reconsider qq = q as Element of the carrier of K;
       take qq;
       thus qq divides p by A1;
       thus 1 <= deg qq by A1,A11,T8,NAT_1:14;
       thus deg qq < deg p by A3,A7,XXREAL_0:1;
       end;
     end;
   end;
now assume AS: p = 0_.(R) or p is Unit of Polynom-Ring R or
       ex q being Element of the carrier of Polynom-Ring R
                                 st q divides p & 1 <= deg q & deg q < deg p;
  per cases by AS;
  suppose p = 0_.(R);
    then p = 0.K by POLYNOM3:def 10;
    hence p is reducible;
    end;
  suppose p is Unit of Polynom-Ring R;
    hence p is reducible;
    end;
  suppose ex q being Element of the carrier of Polynom-Ring R
                                 st q divides p & 1 <= deg q & deg q < deg p;
    hence p is reducible by T88a;
    end;
  end;
hence thesis by A;
end;
