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Let $n$ be a positive integer, and let $\mathbb F$ be a field of positive characteristic $p$ with $\gcd(n,p) = 1$. Where can I find some proofs that the group of all $n$-th roots of unity (in an algebraic closure of $\mathbb F$) form a cyclic group? Would you be so kind to provide an account of all known ones of which you are aware (together with references thereof)? Thank you in advance.

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    When $\gcd(n,p) \neq 1$ they still form a cyclic group! Just not of order $n$ anymore, i.e. when $n=p$ the $n^{th}$ roots of unity is the trivial group.2012-12-24
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    Your request is somewhat odd; what exactly are you trying to do?2012-12-24

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