Given a monic irreducible polynomial $f\in\mathbb{Z}[x]$, I'd like to know for how many primes p we have that $f \bmod p$ is irreducible.
In the link: How many primes stay inert in a finite (non-cyclic) extension of number fields?, the analysis gives rise to a characterization in the case $\mathbb{Q}[x]/(f)$ is Galois over $\mathbb{Q}$.
Now I am wondering, is there any characterization known for the non-Galois case?
Let $K$ be a splitting field of $f$ over $\mathbb{Q}$. Since $f$ is irreducible, it is in particular separable; let $\theta_1$, $\theta_2$, ..., $\theta_n$ be the roots of $f$ in $K$. Then $G:=\text{Gal}(K/\mathbb{Q})$ permutes the $\theta_i$, and we can thus think of $G$ as a subgroup of $S_n$. (In fact, everything I say here is also true for reducible, but separable, polynomials, such as $(x^2-2)(x^2-3)(x^2-6)$.)
Let $p$ be an unramified prime for $K$. Then $p$ has an associated Frobenius conjugacy class $\text{Frob}(p)$ in $G$. The irreducible factors of $f(x) \bmod p$ are in bijection with the cycles of $\text{Frob}(p)$, and the degrees of the irreducible factors are equal to the lengths of the cycles. In particular, $f(x)$ is irreducible modulo $p$ if and only if $\text{Frob}(p)$ is an $n$-cycle.
By the Cebotarov density theorem, the fraction of $p$ for which this occurs is $\#(\text{$n$-cycles in $G$})/\#(G)$.
I wrote this up in a blogpost years ago. I learned it from Janusz but, for some reason, Janusz wrote everything in terms of a group $G$ and an index $n$ subgroup $H$ (the stabilizer of $\theta_1$), rather than describing $G$ as a subgroup of $S_n$. I'm curious to hear what sources others would recommend.
This is a supplement to David E Speyer's excellent answer. It was already proved by Frobenius (1880) that the fraction of $p$ for which $f(x)\bmod p$ has a given decomposition type $(n_1,\dotsc,n_t)$ is equal to the proportion of elements of $G$ with cycle pattern $(n_1,\dotsc,n_t)$. This preceded and motivated the celebrated work of Chebotarev (1923). In fact the Chebotarev density theorem was conjectured by Frobenius (1896), as made clear by Chebotarev in the introduction of his paper (both in the Russian and the German version). A good source for this material is Lenstra-Stevenhagen: Chebotarev and his density theorem.
