Klaška, Jiří; Skula, Ladislav On the factorizations of cubic polynomials with the same discriminant modulo a prime. (English) Zbl 1481.11113 Math. Slovaca 68, No. 5, 987-1000 (2018). Summary: Let \(D\in\mathbb Z\) and let \(C_D\) 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 \(C_D\) have the same type of factorization over the Galois field \(\mathbb F_p\). Cited in 2 Documents MSC: 11T06 Polynomials over finite fields Keywords:factorizations of cubic polynomials; Galois fields PDFBibTeX XMLCite \textit{J. Klaška} and \textit{L. Skula}, Math. Slovaca 68, No. 5, 987--1000 (2018; Zbl 1481.11113) Full Text: DOI References: [1] Klaška, J.—Skula, L.: Mordell’s equation and the Tribonacci family, Fibonacci Quart. 49(4) (2011), 310-319. · Zbl 1261.11034 [2] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the real case, Util. Math. 102 (2017), 39-50. · Zbl 1404.11135 [3] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the imaginary case, Util. Math. 103 (2017), 99-109 · Zbl 1404.11136 [4] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the case of discriminants divisible by three, Math. Slovaca 66(4) (2016), 1019-1027. · Zbl 1399.11178 [5] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the case of primes 2 and 3, Math. Slovaca 67(1) (2017), 71-82. · Zbl 1399.11179 [6] Kučera, R.: Revealing two cubic non-residues in a quadratic field locally, Math. Slovaca 68(1) (2018), 53-56. · Zbl 1473.11195 [7] Stickelberger, L.: Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper, Verhand. I. Internat. Math. Kongress (1897), 182-193. · JFM 29.0172.03 [8] Voronoï, G.: On Integral Algebraic Numbers Depending on a Root of an Irreducible Equation of the Third Degree, Master’s dissertation (in Russian), 1894. [9] Voronoï, G.: Sur une propriété du discriminant des fonctions enti‘eres, Verhand. III. Internat. Math. Kongress (1905), 186-189. [10] Ward, M.: The characteristic number of a sequences of integers satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), 153-165. · JFM 57.0182.02 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.