= Root of unity over a finite field
{title2=$E(n,q)=\{a\in\overline{\mathbb F}_q:a^n=1\}$}
= Roots of unity over a finite field
{synonym}
These roots are taken in an <algebraic closure>, not merely in $\mathbb F_q$. If the <characteristic of a field> is $p$ and $n=p^a m$ with $p\nmid m$, then $X^n-1=(X^m-1)^{p^a}$; hence there are $m$ distinct roots. They form a <cyclic group> by the <finite multiplicative subgroup of a field is cyclic> theorem. In particular they have order $n$ when $\gcd(n,q)=1$.
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