Let $f \in \mathbb{Z}[x]$ be irreducible, and let $\bar{f} \in \mathbb{F}_{p}[x]$ be the image of $f$ in the polynomial ring over the finite field with $p$ elements. Is there a general procedure, given $f$ to find the primes $p$ such that $\bar{f}$ is irreducible over $\mathbb{F}_{p}$?
More specifically, the polynomial I am interested in is the 'Fibonacci polynomial' $\phi(x) = x^2 - x - 1$. For which primes is $\bar{\phi}$ irreducible over $\mathbb{F}_{p}$?