Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}\_p[x].$ Let $\rho\_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}\_p$, and let $\rho$ be the number of distinct roots of $f$ in $\mathbb{C}$. Do you have a reference for showing that $\rho\_p = \rho$ for infinitely many primes $p$?




---


My background to this question: I actually use this result to calculate the precise maximal subgroup growth of groups of the type $\mathbb{Z}^k \rtimes \mathbb{Z}$ (and of similar metabelian groups). 



Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?
Yes. Chebotareff (really Frobenius here) density theorem says that each cycle type in the Galois group of $f$ occurs as the splitting type of $f$ modulo $p$ infinitely often. Your case corresponds to the cycle type of the identity element. The Chebotareff theorem says that this happens asymptotically once in $|Gal(f)|$ primes (the statement about infinite number of primes is actually much easier, and follows from the Chinese remainder theorem, but what the heck).


