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

Abstract

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 F p $\Bbb F_p$ .