How to show that there are infinitely many prime numbers $p$ such that the polynomial f has a zero in $\mathbb Z_p$?

Question

Let $f\in \mathbb Z[X]$ be a polynomial of positive degree.How to show that there are infinitely many prime numbers $p$ such that the polynomial $f$ has a zero in $\mathbb Z/p \mathbb Z$ ?

I have no idea of how to start even.

Need help

Answer 1

Hint: Assume there are finitely many primes that divide any values of $f$. Then, for all $n$,

$$f(n)=\pm p_1^{e_1}\cdots p_k^{e_k}$$

for some nonnegative integers $e_1,\cdots,e_k$.

If you look at the range $[-N,N]$ for some large $N$, what proportion of numbers in that range are values of $f$ (asymptotically)? What proportion of those numbers can be written as $\pm p_1^{e_1}\cdots p_k^{e_k}$?