 Root of unity over a finite field

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 $\gcd (n,q)=1$.