Number of factors of irreducible polynomial of degree $mn$ over $\mathbb{F}_p$ in $\mathbb{F}_{p^n}[x].$

Question

Given a polynomial $f(x)$ of degree $mn$ over $\mathbb{F}_p$, how can we calculate how many irreducible factors it has in $\mathbb{F}_{p^n}[x]$?

Since we have $\mathbb{F}_p\subset \mathbb{F}_{p^n}\subset\mathbb{F}_{p^{mn}},$ we must have a factor of degree $m$. But how can we continue this process? I am pretty sure $f(x)$ should factor into a product of polynomial of degree $m$, but don't see an immediate solution.

Answer 1

$$f=\prod_{k=1}^{mn} (x-a^{p^k}) =\prod_{l=1}^n \prod_{k=1}^m (x-(a^{p^l})^{p^{kn}})$$