Let $K$ be a finite field with $p^n$ elements. Then the group of unit is cyclic of order $p^n-1$. What are the primes which are dividing $p^n-1$? What are the Sylow-subgroups of the group of units of $K$? Of course they are cyclic but of which order?



Sylow-subgroups of the group of units of a finite field
What kind of answer do you expect? Of course, not much about the prime factorization of $p^n-1$ can be said, for otherwise one could settle questions about Mersenne primes $2^q-1\in\mathbb P$ ($q$ necessarily a prime). One could even settle the existence question of Fermat primes $F\_k=2^{2^k}+1$, if we knew the prime factorization of $\prod\_{k=0}^nF\_k=2^{2^{n+1}}-1$.


As to your general question, note that $p^n-1=\prod\_{k\mid n}\Phi\_k(p)$, where $\Phi\_k$ is the $k$-th cylotomic polynomial. So one gets some little information about prime factors, as each prime divisor $q$ of $\Phi\_k(p)$ divides $k$ or fulfills $q\equiv1\pmod{k}$.


