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

The key point is to understand the field Q(w) for w a primitive kth root of unity. Call this field Q_{k}. In particular, you want to know that Q_{4n} \neq Q_{2n}. The key fact here is that the field extension Q_{k}/Q has degree phi(k), where phi(k) is the Euler phi function, and phi(4k) \neq phi(2k). For a proof that Q_{k}/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. 


If w is a primitive 2nth root, then the answer is "none". If w is not primitive, then Q(w) has a square root of w if and only if and odd power of w is 1. 

