Suppose w^(2n)=1 (w is a complex number). For which n (if any) \sqrt(w) \in Q(w) ?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
1
|
||||||||||||
|
|
3
|
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. |
|||
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
1
|
If w is a primitive 2n-th 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. |
|||
|
