
theorem baseZ:
{1, zeta} is Basis of VecSp(FAdj(F_Rat,{zeta}),F_Rat)
proof
set F = F_Rat;
B: now let o be object;
   assume o in Base zeta; then
   consider n being Element of NAT such that
   B1: o = zeta|^n & n < deg MinPoly(zeta,F);
   n < 1 + 1 by LL,B1,minpzeta; then
   n <= 1 by NAT_1:13; then
   per cases by NAT_1:25;
   suppose n = 0;
    then o = 1_F_Complex by B1,BINOM:8;
    then o = 1 by COMPLEX1:def 4,COMPLFLD:def 1;
    hence o in {1, zeta} by TARSKI:def 2;
    end;
   suppose n = 1;
    then o = zeta by B1,BINOM:8;
    hence o in {1, zeta} by TARSKI:def 2;
    end;
   end;
now let o be object;
  assume o in {1, zeta}; then
  per cases by TARSKI:def 2;
  suppose o = 1;
    then o = 1_F_Complex by COMPLEX1:def 4,COMPLFLD:def 1
          .= zeta|^0 by BINOM:8;
    hence o in Base zeta by LL,minpzeta;
    end;
  suppose o = zeta;
    then o = zeta|^1 by BINOM:8;
    hence o in Base zeta by LL,minpzeta;
    end;
  end;
then Base zeta = {1, zeta} by B,TARSKI:2;
hence thesis by FIELD_6:65;
end;
