The order of the Galois group of a cyclotomic field over a finite prime field 
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For what $(n,k)$ there exists a polynomial $p(x) \in F_2\[x\]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$?
Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} / (n) )^*$ 

Let $p$ be a prime number.
Let $F = \mathbb{Z}/p\mathbb{Z}$.
Let $l$ be an odd prime number such that $l \neq p$.
Let $X^l - 1 \in F[X]$.
Let $K$ be the splitting field of $X^l - 1$.
Can we determine the degree $K/F$? 
This is a related question.

Let $f$ be the smallest positive integer such that $p^f \equiv 1$ (mod $l$).
Let $\Omega$ be the algebraic closure of $F$.
Let $\omega \neq 1$ be a root of $X^l - 1$ in $\Omega$.
Then, by my answer to this question, the minimal polynomial of $\omega$ over $F$ has degree $f$. Since $F(\omega)$ is the splitting field of $X^l - 1$, The degree of $K/F$ is $f$.
