
theorem Ysplit:
FAdj(F_Rat,{zeta}) is SplittingField of X^3-1
proof
set F = FAdj(F_Rat,{zeta});
Roots(F_Complex,X^3-1) c= the carrier of F
   proof
   A1: {zeta} is Subset of F by FIELD_6:35;
   zeta in {zeta} by TARSKI:def 1; then
   reconsider b = zeta as Element of F by A1;
   A3: F is Subring of F_Complex by FIELD_5:12;
   A4: zeta^2 = zeta * zeta by O_RING_1:def 1
             .= b * b by A3,FIELD_6:16;
   now let o be object;
     assume o in Roots(F_Complex,X^3-1); then
     per cases by lemroots3,ENUMSET1:def 1;
     suppose o = 1;
       then o = 1.F_Complex by COMPLEX1:def 4,COMPLFLD:def 1
             .= 1.F by EC_PF_1:def 1;
       hence o in the carrier of F;
       end;
     suppose o = zeta;
       then o = b;
       hence o in the carrier of F;
       end;
     suppose o = zeta^2;
       hence o in the carrier of F by A4;
       end;
     end;
   hence thesis;
   end; then
B: X^3-1 splits_in F by LLsplit,FIELD_8:27;
now let E be FieldExtension of F_Rat;
  assume C: X^3-1 splits_in E & E is Subfield of F; then
  E: E is Subfield of F_Complex by EC_PF_1:5;
  D: F_Rat is Subfield of E by FIELD_4:7;
  {zeta} is Subset of E
    proof
    F_Complex is E-extending by E,FIELD_4:7; then
    A1: Roots(F_Complex,X^3-1) c= the carrier of E by LLsplit,C,FIELD_8:27;
    zeta in Roots(F_Complex,X^3-1) by lemroots3,ENUMSET1:def 1; then
    zeta in the carrier of E by A1;
    then {zeta} c= the carrier of E by TARSKI:def 1;
    hence thesis;
    end;
  then F is Subfield of E by D,E,FIELD_6:37;
  hence E == F by C,FIELD_7:def 2;
  end;
hence F is SplittingField of X^3-1 by B,FIELD_8:def 1;
end;
