If $|K| = q$ and $(n,q) = 1$ and $F$ is a splitting field of $x^n - 1$ over $K$, then $[F:K]$ is the least positive integer $k$ such that $n \mid q^k - 1$.
My idea: Since $K$ is finite, we know that $q = p^r$ for some prime number $p$ and natural number $r$. Now let $m = [F:K]$, and I am trying to prove that $n$ divides $p^{rm} - 1$. It is well known that $F$ is also the splitting field of the polynomial $x^{p^{rm}} - x$ over $\mathbb{Z}_p$ (so over $K$). Furthermore, $F$ is a simple extension of $\mathbb{Z}_p$. I collected all this information but I am not sure how to conclude my argument. Hints?