How to find the minimal polynomial of a polynomial over a finite field

Question

Let $\mathbb{F}_{p^n}$ be a finite field and $f(x) \in \mathbb{F}_{p^n}[x]$ which is monic and non-constant. My question is:

$i)$ Is there the minimal polynomial $g(x) \in \mathbb{F}_{p}[x] \subset \mathbb{F}_{p^n}[x]$ divisible by $f(x)$ over $\mathbb{F}_{p^m}$ for some $m \vdots n$ ? If there is, is it irreducible over $\mathbb{F}_{p^n}$ ?

$ii)$ If there is, how to find that $g(x)$ ? Is there a (not necessarily feasible) algorithm for that?

Thanks in advance.

Answer 1

Each root of $f$ (as elements in the algebraic closure of $\Bbb F_p$) has a minimal polynomial over $\Bbb F_p$. Multiply each distinct such polynomial, and you have your $g$.

This $g$ will be divisible by $f$ as elements of $\Bbb F_{p^n}[x]$. As such, $g$ is irreducible in $\Bbb F_{p^n}[x]$ iff $f$ is.