
theorem ns2:
not sqrt 2 is Element of FAdj(F_Rat,{zeta})
proof
set F = FAdj(F_Rat,{zeta}), V = VecSp(F,F_Rat);
A: the carrier of F = the carrier of V by FIELD_4:def 6;
{zeta} is Subset of F & zeta in {zeta} by FIELD_6:35,TARSKI:def 1; then
reconsider e = 1.F_Complex, z = zeta as Element of V by A,FIELD_6:def 6;
e = 1 by COMPLEX1:def 4,COMPLFLD:def 1; then
C: Lin{e,z} = the ModuleStr of V by baseZ,VECTSP_7:def 3;
now assume sqrt 2 in the carrier of FAdj(F_Rat,{zeta}); then
  sqrt 2 in Lin{e,z} by FIELD_4:def 6,C; then
  consider l being Linear_Combination of {e,z} such that
  C1: sqrt 2 = Sum l by VECTSP_7:7;
      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 a = l.e, b = l.z as Element of F_Complex;
  C2: sqrt 2 = a + b * zeta
      proof
      E: l.e * e = a
         proof
         E0: F_Rat is Subfield of F by FIELD_6:36;
         1.F_Complex = 1.F by EC_PF_1:def 1; then
         E1: [l.e,e] in [:the carrier of F_Rat,the carrier of F:]
             by ZFMISC_1:def 2;
             the carrier of F_Rat c= the carrier of F by E0,EC_PF_1:def 1; then
         E2: [:the carrier of F_Rat,the carrier of F:]  c=
             [:the carrier of F,the carrier of F:] by ZFMISC_1:96;
         thus l.e * e
        = ((the multF of F)|[:the carrier of F_Rat,the carrier of F:]).(l.e,e)
          by FIELD_4:def 6
       .= (the multF of F).(l.e,e) by E1,FUNCT_1:49
       .= ((the multF of F_Complex)||(the carrier of F)).(l.e,e)
          by EC_PF_1:def 1
       .= a * 1.F_Complex by E2,E1,FUNCT_1:49 .= a;
         end;
      F: l.z * z = b * zeta
         proof
         F0: F_Rat is Subfield of F by FIELD_6:36;
         {zeta} is Subset of F & zeta in {zeta} by FIELD_6:35,TARSKI:def 1;
         then
         F1: [l.z,z] in [:the carrier of F_Rat,the carrier of F:]
             by ZFMISC_1:def 2;
             the carrier of F_Rat c= the carrier of F by F0,EC_PF_1:def 1; then
         F2: [:the carrier of F_Rat,the carrier of F:]  c=
             [:the carrier of F,the carrier of F:] by ZFMISC_1:96;
         thus l.z * z
        = ((the multF of F)|[:the carrier of F_Rat,the carrier of F:]).(l.z,z)
          by FIELD_4:def 6
       .= (the multF of F).(l.z,z) by F1,FUNCT_1:49
       .= ((the multF of F_Complex)||(the carrier of F)).(l.z,z)
          by EC_PF_1:def 1
       .= b * zeta by F2,F1,FUNCT_1:49;
         end;
       a in F & b * zeta in F
           proof
           F_Rat is Subfield of F by FIELD_6:36; then
           G2: the carrier of F_Rat c= the carrier of F by EC_PF_1:def 1;
           hence a in F;
           G3: F is Subring of F_Complex by FIELD_5:12;
           {zeta} is Subset of F & zeta in {zeta} by TARSKI:def 1,FIELD_6:35;
           then reconsider u = zeta, v = b as Element of F by G2;
           v * u = b * zeta by G3,FIELD_6:16;
           hence b * zeta in F;
           end; then
      G: [a,b*zeta] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
      l.e * e + l.z * z
          = (the addF of F).(a,b*zeta) by E,F,FIELD_4:def 6
         .= ((the addF of F_Complex)||(the carrier of F)).(a,b*zeta)
            by EC_PF_1:def 1
         .= a + b * zeta by G,FUNCT_1:49;
      hence thesis by C1,VECTSP_6:18;
      end;
  C4: sqrt 2 = (a + b * (-1/2)) + <i> * (b * (sqrt 3) / 2) by C2; then
      Im(a + b * zeta) = b * (sqrt 3) / 2 by COMPLEX1:12; then
      b * (sqrt 3) / 2 = 0 by C2;
  hence contradiction by C4,INT_2:28,IRRAT_1:1;
  end;
hence thesis;
end;
