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.