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Given positive integer $n$, an element $\alpha$ of a field is an nth root of unity if $\alpha^n=1$. It is a primitive nth root of unity if $\alpha^n=1$ and $\alpha^m\neq1$ for $0<m<n$.

I've seen this definition of primitive roots of unity for fields, and even for rings with unity, but not for groups. Would the definition make sense, and if not, why not?

Remark I suppose if this definition makes sense, the torsion subgroup would precisely contain the nth roots of unity for each $n$?

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    $\begingroup$ For groups, aren't these just the elements of order $n$? $\endgroup$
    – Misha Lavrov
    Nov 23, 2017 at 1:38
  • $\begingroup$ Yes, if primitive. If not, I imagine we could just characterize elements which are $n$th root of unity as those "elements whose order divides n"? Which seems a bit of a mouthful. Does anybody know any texts which use the phrase "roots of unity" (or another term) in this context? $\endgroup$
    – SSF
    Nov 23, 2017 at 1:42

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In a group, we already have a name for this: we call it an element of order $n$. In a ring, it's helpful to be more specific -- saying "$x$ has order $n$'' might mean that $n\cdot x=0$ or that $x^n=1$.

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  • $\begingroup$ @Misha: you beat me to it -- but I had to answer, because I don't have enough rep to comment. $\endgroup$
    – swensonj
    Nov 23, 2017 at 1:41
  • $\begingroup$ Yes, thanks. I should have just asked about "roots of unity" and omitted "primitive". Then there might be reason to use the term "$n$th roots of unity" for those elements whose order divides n. $\endgroup$
    – SSF
    Nov 23, 2017 at 1:48
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    $\begingroup$ @swensonj: it is good to have the answers as answers rather than hidden away in comments anyway. Welcome to MSE! $\endgroup$
    – hmakholm left over Monica
    Nov 23, 2017 at 1:49

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