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

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

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

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

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