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

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is it homework ? – David Lehavi Oct 22 '09 at 7:38
Perhaps the title should say something about the field Q(w)... how about "Is root of unity w a square in Q(w)?" – Ilya Nikokoshev Oct 22 '09 at 8:01
This could be homework for a second course in number theory, but it also could be a lemma needed by someone in a field far from number theory. I would leave this open. – David Speyer Oct 22 '09 at 10:44
up vote 3 down vote accepted

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.

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Ok, thanks. Yes I need this for some lemma. – user966 Oct 22 '09 at 12:16

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.

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I want w to be a primitive 2n-th root. Can you give me a hint of how to prove the fact that you mentioned? – user966 Oct 22 '09 at 6:59

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