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For which $n,m$ is $\Bbb Q(w)\subseteq \Bbb Q(\zeta)$, where $w$ and $\zeta$ are primitive $n$ and $m$th roots of unity respectively.

And, what are the cyclotomic extensions containing $i$?

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Have you tried any examples? – Zev Chonoles Apr 28 '12 at 7:15
I've changed your choice of letter p to the Greek letter $\zeta$, because far too often the former is understood to be a prime number, especially in the context of cyclotomy. I hope you don't mind the stylistic change. – anon Apr 28 '12 at 7:15

the answer is :"if and only if n|m or n|2m with m odd"

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