Let $n \geq 1$ be an integer, and let $g(x) \in \mathbb{F}_p[x]$ be monic and irreducible. What is the $p$-adic density in $\mathbb{Z}_p[x]$ of monic irreducible polynomials $f(x)$ such that $f(x) \equiv g(x)^n \pmod p$?
Here are some situations in which it isn't hard to at least get a bound on this density:
- When $n = 1$, Hensel's lemma states that every lift of $g(x)$ to $\mathbb{Z}_p[x]$ is irreducible.
- When $g(x) = x$, the set of monic irreducible lifts of $g(x)^n$ to $\mathbb{Z}_p[x]$ contains the degree-$n$ Eisenstein polynomials.
Is it possible to compute this $p$-adic density in any more general setting?