Dear all,

I am looking for a proof or a reference of the following statement:

Let $f$ be a non-constant polynomial with integer coefficients. Then the sum $\sum \{1/p \mid f \text{ has a root modulo } p\}$ diverges.

I am pretty sure that I saw it somewhere before but I cannot remember and I failed to find it in number theory books. A possible routes that has already been suggested to me is actually showing that the sums of reciprocals for which f even decomposes into linear factors has positive density which should stem from Galois theory. I am however not an expert in Galois theory so that I would prefer a direct proofs or a reference.

Thanks in advance, Alberto