It is shown that the splitting modulo a prime p of a given monic, integral, irreducible cubic with non-square discriminant is equivalent to p being represented by forms in a certain subgroup of index 3 in the form class group of discriminant equal to the discriminant of the field defined by the cubic.

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Journal Journal of the London Mathematical Society
Citation
Spearman, B.K. (Blair K.), & Williams, K.S. (2001). The cubic congruence x3+Ax2 + Bx + C = 0 (mod p) and binary quadratic forms II. Journal of the London Mathematical Society, 64(2), 273–274.