From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented:
Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If $f$ is irreducible over the integers, and $\sqrt[p]{(-1)^{\deg(f)} f(0)}$ is irrational, then $f(x^p)$ is also irreducible over the integers.
I've reproduced the proof here. I'd like to see an elementary proof that does not rely on field theory, if possible.
