TITLE: For an irreducible monic $f\in\mathbb{Z}[x]$, is it true that $\overline{f}\in\mathbb{F}_p[x]$ will be irreducible for all but finitely many $p$?
QUESTION [5 upvotes]: Let $f\in\mathbb{Z}[x]$ be an irreducible monic, then it need not be the case that $\overline{f}\in\mathbb{F}_p[x]$ will be irreducible. But will it be irreducible for all but finitely many $p$? My intuition is that this is probably false, but I don't have any idea of how to find a counterexample.

REPLY [5 votes]: A result in the opposite direction of the provided answer is this: if $ f \in \mathbf Z[X] $ is irreducible monic, then there are infinitely many primes $ p $ such that modulo $ p $, $ f $ splits into linear factors. This follows from the general result that infinitely many primes split completely in any given number field; coupled with the fact that for all but finitely many primes, splitting of $ f $ modulo $ p $ mirrors the splitting of $ p $ in the ring of integers of $ \mathbf Q(a) $, where $ a $ is a root of $ f $. Therefore, there is in fact no $ f $ (of degree greater than $ 1 $) which makes the statement of the question work.