On the factorizations of cubic polynomials with the same discriminant modulo a prime

Abstract

Let D ∈ ℤ and let CD be the set of all monic cubic polynomials with integer coefficients having a discriminant equal to D. In this paper, we devise a general method of establishing whether, for a prime p, all polynomials in CD have the same type of factorization over the Galois field $\Bbb F_p$ .

On cubic polynomials with a given discriminant

“…Creating tables of these representatives and analyzing them can lead to the discovery of new interesting facts. These tables can also be useful for a better understanding of the law of inertia for the factorization of cubic polynomials, which has been studied in [11][12][13][14][15][16].…”