
theorem baszeta:
{ 1, zeta } is Basis of
VecSp(FAdj(F_Rat,{3-CRoot(2),zeta}), FAdj(F_Rat,{3-CRoot(2)}))
proof
set F = FAdj(F_Rat,{3-CRoot(2), zeta}), K = FAdj(F_Rat,{3-CRoot(2)});
C: FAdj(F_Rat,{3-CRoot(2), zeta})
     = FAdj(F_Rat,{3-CRoot(2)}\/{zeta}) by ENUMSET1:1
    .= FAdj(K,{zeta}) by FIELD_7:35;
reconsider z = zeta as K-algebraic Element of F_Complex;
H: deg MinPoly(z,K) = 2 by mmm,LL,FIELD_4:20;
B: now let o be object;
   assume o in Base z; then
   consider n being Element of NAT such that
   B1: o = z|^n & n < deg MinPoly(z,K);
   n < 1 + 1 by mmm,LL,FIELD_4:20,B1; 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, z} by TARSKI:def 2;
    end;
   suppose n = 1;
    then o = z by B1,BINOM:8;
    hence o in {1, z} by TARSKI:def 2;
    end;
   end;
now let o be object;
  assume o in {1, z}; then
  per cases by TARSKI:def 2;
  suppose o = 1;
    then o = 1_F_Complex by COMPLEX1:def 4,COMPLFLD:def 1
          .= z|^0 by BINOM:8;
    hence o in Base z by H;
    end;
  suppose o = z;
    then o = z|^1 by BINOM:8;
    hence o in Base z by H;
    end;
  end; then
Base z = {1, z} by B,TARSKI:2;
hence thesis by C,FIELD_6:65;
end;
