
theorem 32split:
FAdj(F_Rat,{3-CRoot(2), zeta}) is SplittingField of X^3-2
proof
set F = FAdj(F_Rat,{3-CRoot(2), zeta});
Roots(F_Complex,X^3-2) c= the carrier of F
   proof
   A1: {3-CRoot(2),zeta} is Subset of F by FIELD_6:35;
   3-Root(2) in {3-CRoot(2), zeta} &
   zeta in {3-CRoot(2), zeta} by TARSKI:def 2; then
   reconsider a = 3-Root(2), b = zeta as Element of F by A1;
   A3: F is Subring of F_Complex by FIELD_4:def 1;
   A5: zeta * zeta = b * b by A3,FIELD_6:16;
   now let o be object;
     assume o in Roots(F_Complex,X^3-2); then
     per cases by lemroots,ENUMSET1:def 1;
     suppose o = 3-Root(2);
       then o = a;
       hence o in the carrier of F;
       end;
     suppose o = 3-Root(2) * zeta;
       then o = 3-CRoot(2) * zeta
             .= a * b by A3,FIELD_6:16;
       hence o in the carrier of F;
       end;
     suppose o = 3-Root(2) * zeta^2;
       then o = 3-CRoot(2) * (zeta * zeta) by O_RING_1:def 1
             .= a * (b * b) by A3,A5,FIELD_6:16;
       hence o in the carrier of F;
       end;
     end;
   hence thesis;
   end; then
B: X^3-2 splits_in F by LLsplit,FIELD_8:27;
now let E be FieldExtension of F_Rat;
  assume C: X^3-2 splits_in E & E is Subfield of F; then
  E: E is Subfield of F_Complex by EC_PF_1:5; then
  F: E is Subring of F_Complex by FIELD_5:12;
  D: F_Rat is Subfield of E by FIELD_4:7;
  {3-CRoot(2),zeta} is Subset of E
    proof
    F_Complex is E-extending by E,FIELD_4:7; then
    A1: Roots(F_Complex,X^3-2) c= the carrier of E by LLsplit,C,FIELD_8:27;
    3-Root(2) in Roots(F_Complex,X^3-2) &
    3-CRoot(2) * zeta in Roots(F_Complex,X^3-2)
          by lemroots,ENUMSET1:def 1; then
    reconsider a = 3-Root(2),
               b = 3-CRoot(2) * zeta as Element of E by A1;
    A3: 3-CRoot(2) is non zero;
    a" = 3-CRoot(2)" by E,FIELD_6:18; then
    A4: a" * b = 3-CRoot(2)" * (3-CRoot(2) * zeta) by F,FIELD_6:16
              .= (3-CRoot(2)" * 3-CRoot(2)) * zeta
              .= 1.F_Complex * zeta by A3,VECTSP_2:9;
    now let o be object;
      assume o in {3-CRoot(2),zeta}; then
      per cases by TARSKI:def 2;
      suppose o = 3-CRoot(2); then
        o = a;
        hence o in the carrier of E;
        end;
      suppose o = zeta;
        hence o in the carrier of E by A4;
        end;
      end;
    then {3-CRoot(2),zeta} c= the carrier of E;
    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^3-2 by B,FIELD_8:def 1;
end;
