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Show that the primitive n-th roots of unity have the form $e^{2ki\pi/n}$ for $k,n$ coprime for $0\leq k\leq n$.

Since all primitive n-th roots of unity are n-th roots of unity by definition they all have that form, the question is, how to show $k$ and $n$ are coprime.

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    Put another way: if $k$ and $n$ are _not_ coprime, then is $e^{2\pi i k/n}$ a primitive $n$-th root of $1$? What does being a primitive $n$-th root mean?2012-01-21
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    please say the definition of primitive root. thanks.2013-09-27

3 Answers 3