Let $f(x)=x^3+3x+12$. Now if we have the relation $$f(x)\equiv0\pmod p$$ for some prime $p$, then what are the values of $p$ for which this equation is solvable for $x$?
I know that the cubic reciprocity law tells us when we have the solution of a congruence of the form $X^3\equiv a \pmod p$ for some prime $p$ but for this form of cubic I dont know any result... thanks in advance for any help.
The Galois group of this polynomial is non-abelian, which implies that there is no straightforward description of the primes $p$ for which there is a root modulo $p$ in terms of congruences of $p$ modulo fixed integers. (If the Galois group is abelian, then this set can be defined in terms of congruences.) There is a result (Chebyshev?) that tells you the asymptotic proportion of primes with this property.
For a cubic polynomial with Galois group $S_3$, I think there is one root in ${\mathbb F}_p$ for roughly half of the primes, three roots for $1/6$ of the primes, and no roots for $1/3$ of them.