- 509
- 324
Suppose [itex]\alpha[/itex] is [itex]n[/itex]-th root of unity and [itex]\alpha \in \mathbb F_{q^m}[/itex] for some [itex]m\in\mathbb N[/itex]. Then by Lagrange [itex]n \mid q^m-1[/itex]. It would suffice to compute the order of [itex]q[/itex] modulo [itex]n[/itex]. As [itex](q,n)=1[/itex] it holds that [itex]n\mid q^{\phi (n)} -1[/itex], where [itex]\phi[/itex] is the Euler totient map. So the order of [itex]q[/itex] must be a divisor of [itex]\phi (n)[/itex].
Not sure how to explicitly compute [itex]m[/itex].
Not sure how to explicitly compute [itex]m[/itex].