
theorem
for F being polynomial_disjoint Field,
    p being irreducible Element of the carrier of Polynom-Ring F
holds embField(canHomP p) is polynomial_disjoint iff p is linear
proof
let F be polynomial_disjoint Field,
    p be irreducible Element of the carrier of Polynom-Ring F;
set FP = Polynom-Ring p, K = embField(canHomP p), X = <%0.F,1.F%>;
X: [#] F = the carrier of F &
   [#] (Polynom-Ring p) = the carrier of Polynom-Ring p;
A: now assume AS: deg p > 1;
   H: the carrier of FP = {q where q is Polynomial of F : deg q < deg p}
      by RING_4:def 8;
   len X = 2 by POLYNOM5:40; then
   D: deg X = 2 - 1 by HURWITZ:def 2; then
   C: X in the carrier of FP by H,AS;
   now assume X in rng(canHomP p);
     then consider o being object such that
     B1: o in dom(canHomP p) & (canHomP p).o = X by FUNCT_1:def 3;
     reconsider a = o as Element of F by B1;
     X = a|F by B1,defch;
     hence contradiction by D,RATFUNC1:def 2;
     end;
   then X in (the carrier of FP) \ rng(canHomP p) by C,XBOOLE_0:def 5;
   then X in ((the carrier of FP) \ rng(canHomP p)) \/ (the carrier of F)
     by XBOOLE_0:def 3;
   then X in carr(canHomP p) by X,FIELD_2:def 2; then
   A: X in the carrier of K by FIELD_2:def 7;
   now let n be Element of NAT;
    per cases by NAT_1:23;
    suppose B: n = 0;
      hence <%0.F,1.F%>.n = 0.F by POLYNOM5:38
              .= 0.K by FIELD_2:def 7
              .= <%0.K,1.K%>.n by B,POLYNOM5:38;
      end;
    suppose B: n = 1;
      hence <%0.F,1.F%>.n = 1.F by POLYNOM5:38
              .= 1.K by FIELD_2:def 7
              .= <%0.K,1.K%>.n by B,POLYNOM5:38;
      end;
    suppose B: n >= 2;
      hence <%0.F,1.F%>.n = 0.F by POLYNOM5:38
              .= 0.K by FIELD_2:def 7
              .= <%0.K,1.K%>.n by B,POLYNOM5:38;
      end;
    end;
   then <%0.F,1.F%> = <%0.K,1.K%>;
   then X in the carrier of Polynom-Ring K by POLYNOM3:def 10;
   then X in [#]K /\ [#]Polynom-Ring K by A,XBOOLE_0:def 4;
   hence K is non polynomial_disjoint by FIELD_3:def 3;
   end;
H: 0.K = 0.F by FIELD_2:def 7;
B: now assume C: p is linear;
   then D: [#]Polynom-Ring K = [#]Polynom-Ring F by H,lempk,polyd;
   [#]K = [#]F by C,polyd;
   then [#]K /\ [#]Polynom-Ring K = {} by D,FIELD_3:def 3;
   hence K is polynomial_disjoint by FIELD_3:def 3;
   end;
now assume C: K is polynomial_disjoint;
   deg p >= 0 + 1 by INT_1:7,RING_4:def 4;
   hence p is linear by A,C,XXREAL_0:1;
   end;
hence thesis by B;
end;
