I am studying fields theory and Galois theory as an undergraduate, and as such we are studying finite fields and their properties.
In this case, I was wondering about a common procedure for which I did not have a detailed explanation in my notes.
Let's take the following example:
It is requested to decompose $X^{16}-X$ into $\mathbb{F}_4$ and $\mathbb{F}_8$, where $\mathbb{F}_n$ refers to the Galois field of $n$ elements.
I understand that the difficulty lies in the fact that determining the irreducibility of polynomials in finite fields is not a trivial problem.
The problem can be generalized, for example asking to decompose $X^{(p^n)}-X$ into $\mathbb{F}_{p^m}$ where $p\in\mathbb{N}$ is a prime and $m\leq n$.
I understand that the procedure to be carried out will have to do with decomposition fields and Moore's theorem, but I would like to know if someone can provide a detailed explanation, even if it is for a particular case, or refer me to a source where I can find it.
Greetings and thanks in advance.