
theorem
not F_Complex is SplittingField of X^2+X+1
proof
set F = FAdj(F_Rat,{zeta});
now assume A: F_Complex is SplittingField of X^2+X+1;
  X^2+X+1 splits_in F by Xsplit,FIELD_8:def 1; then
  F_Complex == F by A,FIELD_8:def 1; then
  F_Complex is Subfield of F by FIELD_7:def 2; then
  D: the carrier of F_Complex c= the carrier of F by EC_PF_1:def 1;
  F_Real is Subfield of F_Complex by FIELD_4:7; then
  the carrier of F_Real c= the carrier of F_Complex by EC_PF_1:def 1;
  hence contradiction by D,ns2;
  end;
hence thesis;
end;
