Suppose w^(2n)=1 (w is a complex number).
For which n (if any) \sqrt(w) \in Q(w) ?



Is an nth root of unity a square?
The key point is to understand the field Q(w) for w a primitive kth root of unity. Call this field Qk. In particular, you want to know that Q4n \neq Q2n.


The key fact here is that the field extension Qk/Q has degree phi(k), where phi(k) is the Euler phi function, and phi(4k) \neq phi(2k). For a proof that Qk/Q has degree phi(k), see the early parts of any book on cyclotomic fields. This is probably also done in many Galois theory books but I don't know which ones.


