## Polynomial

Let $f(x)$ be a polynomial in $\mathbb Z[x]$ with leding coefficient 1. Assume that the corresponding polynomial $g(x) \in \mathbb F_p[x]$ has a root in $\mathbb F_p[x]$ for every prime p.

Show that $f(x)$ is reducible in $\mathbb Z[x]$.

My hint says that I should use Dedekind-Frobeniuss Theorem, and the following:

A transitive subgroup of $S_n$ for $n>1$ contains a permutation without fixpoints.

How can I use these things to show the result any suggestions? Thanks.