Does Q(Zeta) = Q(Zeta_12), where Zeta is ANY 12th roots of unity & Zeta_12 = cos(2pi/12)+isin(2pi/12).
In fact, does the general case where we have n instead of 12 hold?
I managed to prove that Q(Zeta) is a subset of Q(Zeta_12) - this follows from the fact that Q(Zeta) is the smallest field with Q & Zeta in it & Zeta = (Zeta_12)^k where k = 1,2,...,12. Therefore Q(Zeta) is a subset of Q(Zeta_12).
But how can I prove that Q(Zeta) is a subset of Q(Zeta_12)? ie - how can I show that Zeta_12 is an element of Q(Zeta) ? :-s
Many thanks x