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 $p$?
If the discriminant of your cubic polynomial is a square $Disc(f) \in \Bbb{Q}^{*2}$ or if $f$ is reducible then it is close to the quadratic case, otherwise it gets much more complicated : the Galois group of the splitting field is $S_3$ and it is the first instance of class field theory (and for negative discriminant elliptic curves with complex multiplication).
Your question becomes the characterization of the non-abelian Artin L-function $L(s,\rho)=\zeta_K(s)/\zeta(s)$ where $K=\Bbb{Q}[x]/(f)$, class field theory will say that $L(s,\rho) = L(s,\psi)$ where $\psi$ is an Hecke character of $F$, relating the factorization of $f\bmod p$ with the norm and ideal class of the primes of $\Bbb{Z}+\Delta O_F$ above $p$.
On the web there are several detailed examples about $x^3+x+1$ but I never recall how to find them.