I'm trying to understand an equation occurring in the proof of Lemma 3 in Adleman and Odlyzko's Irreducibility Testing and Factorization of Polynomials:
Let $f \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $n$, let $D = |\text{disc } f|$ be the absolute value of its discriminant and let $W(p) = |\{x \in \mathbb{Z}/p\mathbb{Z} \mid f(x) = 0\}|$ for any positive prime $p \in \mathbb{N}$. Set \begin{equation} S(x) = \sum_{\substack{0 \leq p \leq x \\ p \text{ prime} \\ p \not| D}} (1 - W(p)), \end{equation} for $x \in \mathbb{R}$. I'd like to understand how to prove the asymptotic bound \begin{equation} |S(x)| = O(\sqrt{x} \log(Dx^n)), \end{equation} which is the second equation (without number) in the proof of the abovementioned lemma. The proof says that this follows from the effective Chebotarev density theorem (see Wikipedia) "since the discriminant of the field generated by a root of $f(x)$ is bounded in absolute value by $D$".
I don't understand what's going on here. What is the extension of $\mathbb{Q}$ we're considering? The field generated by a root of $f$ will in general not be Galois, so it seems to me that the splitting field of $f$ will be involved, but its discriminant is not bounded by $D$. The example given by Jingwei Xue in this thread suggests to me that it will not even be polynomial in $D$ in the worst case.
But even if this obstacle can be overcome, I don't know wich conjugacy class of the Galois group to look at. Maybe the trivial one containing only the identity, because it corresponds to primes that split completely?