Let $f$ be the polynomial $X^3 -rX -s$, where $r>0$ and $s>0$ and let $\Delta$ be its discriminant. Suppose that $\Delta$ is a square, and that $p$ and $q$ are primes such that $\Delta$ divides $p-q$. Under those conditions, is it always the case that $f$ is irreducible in $F_{p}$ iff f is irreducible in $F_{q}$? If $\Delta$ is not a square, it’s easy to find counterexamples.
A question on polynomial irreducibility over finite fields
