
theorem T88:
for F being domRing, p being Element of the carrier of Polynom-Ring F
holds p is Unit of Polynom-Ring F implies deg p = 0
proof
let L be domRing;
let p be Element of the carrier of Polynom-Ring L;
set R = Polynom-Ring L;
H: 1.R = 1_.(L) by POLYNOM3:def 10;
assume AS: p is Unit of R;
   then H0: p <> 0.R;
   then H1: p <> 0_.(L) by POLYNOM3:def 10;
   reconsider degp = deg p as Element of NAT by H0,T8a;
   p divides 1_.(L) by AS,H,GCD_1:def 20;
   then consider t being Polynomial of L such that
   H3: p *' t = 1_.(L) by T2;
   reconsider t as Element of the carrier of R by POLYNOM3:def 10;
   H5: t <> 0_.(L) by POLYNOM3:34,H3;
   then t <> 0.R by POLYNOM3:def 10;
   then reconsider degt = deg t as Element of NAT by T8a;
   degt + degp = deg(1_.(L)) by H1,H3,H5,HURWITZ:23
              .= 1 - 1 by POLYNOM4:4;
   hence deg p = 0;
end;
