This is a rather basic question, but I can't seem to find any reference here on StackExchange. Is it true that, given a polynomial $p(x) \in \Bbb{F}_p[x]$ of degree $n$, we have that $p(x)$ is always reducible in $\Bbb{F}_{p^n}[x]$?
If it is not true, then I'm in particular more interested in the case where $n = 2$.
This is true. If $P(x)$ is irreducible over $\mathbb F_{p^n}$ then it is certainly irreducible over $\mathbb F_{p}$. But that means that $\mathbb F_p(\alpha)$ for some root $\alpha$ of $P$ is a degree $n$ extension of $\mathbb F_p$, and hence equal to $\mathbb F_{p^n}$, and so $P(x)$ would have a root in $\mathbb F_{p^n}$, contradicting the initial assumption that it was irreducible.