
theorem thirr1:
for F being algebraic-closed Field,
    p being Element of the carrier of Polynom-Ring F
holds p is irreducible iff deg p = 1
proof
let R be algebraic-closed Field,
    p be Element of the carrier of Polynom-Ring R;
set K = Polynom-Ring R;
now assume AS: p is irreducible;
  then p is non constant;
  then (len p -1) + 1 > 0 + 1 by XREAL_1:6;
  then consider x being Element of R such that
  A: x is_a_root_of p by POLYNOM5:def 8,POLYNOM5:def 9;
  consider q being Polynomial of R such that
  B: p = rpoly(1,x) *' q by A,HURWITZ:33;
  reconsider y = q, z = rpoly(1,x) as Element of Polynom-Ring R
    by POLYNOM3:def 10;
  p = z * y by B,POLYNOM3:def 10; then z divides p;
  then C: z is Unit of K or z is_associated_to p by AS,RING_2:def 9;
  z is non constant by HURWITZ:27;
  then consider e being Element of K such that
  D: e is unital & z * e = p by C,GCD_1:18;
  reconsider u = e as Element of the carrier of K;
  F: deg u = 0 by D,T88;
  rpoly(1,x) *' u = p by D,POLYNOM3:def 10;
  hence deg p = deg(rpoly(1,x)) + deg u by D,HURWITZ:23
             .= 1 by F,HURWITZ:27;
  end;
hence thesis by thirr;
end;
