
theorem
Roots X^3-1 = {1}
proof
A: now let o be object;
   assume o in {1}; then
   B: o = 1.F_Rat by TARSKI:def 1,GAUSSINT:13;
   eval(X^3-1,1.F_Rat)
       = (1.F_Rat)|^(2+1) - 1.F_Rat by LL31,GAUSSINT:13
      .= ((1.F_Rat)|^2 * (1.F_Rat)|^1) - 1.F_Rat by BINOM:10
      .= ((1.F_Rat)|^(1+1) * 1.F_Rat) - 1.F_Rat by BINOM:8
      .= ((1.F_Rat)|^1 * (1.F_Rat)|^1) - 1.F_Rat by BINOM:10
      .= ((1.F_Rat)|^1 * 1.F_Rat) - 1.F_Rat by BINOM:8
      .= (1.F_Rat * 1.F_Rat) - 1.F_Rat by BINOM:8
      .= 0.F_Rat by RLVECT_1:15; then
   1.F_Rat is_a_root_of X^3-1 by POLYNOM5:def 7;
   hence o in Roots X^3-1 by B,POLYNOM5:def 10;
   end;
H: Roots(F_Complex,X^3-1) =
   {a where a is Element of F_Complex : a is_a_root_of X^3-1,F_Complex}
   by FIELD_4:def 4;
now let o be object;
   assume C: o in Roots X^3-1; then
   reconsider a = o as Element of F_Rat;
   F_Rat is Subfield of F_Complex by FIELD_4:7; then
   the carrier of F_Rat c= the carrier of F_Complex by EC_PF_1:def 1; then
   reconsider c = a as Element of F_Complex;
   G: a is_a_root_of X^3-1 by C,POLYNOM5:def 10;
   0.F_Complex = 0.F_Rat by GAUSSINT:13,COMPLFLD:def 1
              .= eval(X^3-1,a) by G,POLYNOM5:def 7
              .= Ext_eval(X^3-1,c) by FIELD_6:10;
   then c is_a_root_of X^3-1,F_Complex by FIELD_4:def 2;
   then D: a in {1, zeta, zeta^2} by H,lemroots3;
   F_Rat is Subfield of F_Real by FIELD_4:7; then
   E: the carrier of F_Rat c= the carrier of F_Real by EC_PF_1:def 1;
   now assume a <> 1; then
     per cases by D,ENUMSET1:def 1;
     suppose a = zeta;
       hence contradiction by E,FIELD_7:def 5;
       end;
     suppose a = zeta^2;
       hence contradiction by E,FIELD_7:def 5;
       end;
     end;
   hence o in {1} by TARSKI:def 1;
   end;
hence thesis by A,TARSKI:2;
end;
