
theorem
not X^2+X+1 splits_in F_Rat
proof
I: F_Rat is FieldExtension of F_Rat by FIELD_4:6;
   F_Rat is Subfield of F_Real by FIELD_4:7; then
K: the carrier of F_Rat c= the carrier of F_Real by EC_PF_1:def 1;
now assume X^2+X+1 splits_in F_Rat; then
  A: {zeta, zeta^2} c= the carrier of F_Rat by rootz,I,LLsplit,FIELD_8:27;
  zeta in {zeta, zeta^2} by TARSKI:def 2;
  hence contradiction by A,K,FIELD_7:def 5;
  end;
hence thesis;
end;
