Find $n$ such that $f$ is reducible or give a proof.

Question

Let $f = X^4 + 2$ and let $n$ be odd. I am looking for a proof or a counter example to show that $f$ is irreducible or reducible over $\mathbb{F}_{5^n}$.

Answer 1

First show that $X^4+2$ is irreducible over $\mathbb F_5$. Then use the result proved in this answer.