In this paper it is shown that the splitting modulo a prime p of a given monic irreducible cubic polynomial with integral coefficients 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 cubic.

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Persistent URL dx.doi.org/10.1112/jlms/s2-46.3.397
Journal Journal of the London Mathematical Society
Citation
Spearman, B.K. (Blair K.), & Williams, K.S. (1992). The cubic congruence x3 + ax2 + bx + c = 0(mod p) and binary quadratic forms. Journal of the London Mathematical Society, s2-46(3), 397–410. doi:10.1112/jlms/s2-46.3.397