Given a polynomial $f(x) = ax^3 + bx^2 + cx + d \equiv 0 \pmod{p}$, $a \not\equiv 0 \pmod{p}$, I would like to classify all primes $p$ so that there exists $\alpha \in \Bbb{F}_p$ in which $f(\alpha) \equiv 0 \pmod{p}$.
This can be done in the case of quadratic equations, in which quadratic reciprocity is the answer. Is there a similarly simple method of checking if $f(x)$ has a root mod $p$?
from Hot Weekly Questions - Mathematics Stack Exchange
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