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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

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Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2018: 0.29

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1337-2211
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Volume 68, Issue 5

Issues

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

Jiří Klaška
  • Institute of Mathematics Faculty of Mechanical Engineering, Brno University of Technology, Technická, 616 69 Brno, Czech Republic
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/ Ladislav Skula
  • Institute of Mathematics Faculty of Mechanical Engineering, Brno University of Technology, Technická, 616 69 Brno, Czech Republic
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  • Other articles by this author:
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Published Online: 2018-10-20 | DOI: https://doi.org/10.1515/ms-2017-0161

Abstract

Let D 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 Fp.

MSC 2010: Primary 11T06; 11D25; 11D05

Keywords: cubic polynomial; type of factorization; discriminant

References

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    Klaška, J.—Skula, L.: Mordell’s equation and the Tribonacci family, Fibonacci Quart. 49(4) (2011), 310-319.Google Scholar

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    Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the real case, Util. Math. 102 (2017), 39-50.Google Scholar

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    Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials - the imaginary case, Util. Math. 103 (2017), 99-109Google Scholar

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    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.Google Scholar

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    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.Google Scholar

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    Kučera, R.: Revealing two cubic non-residues in a quadratic field locally, Math. Slovaca 68(1) (2018), 53-56.Google Scholar

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    Stickelberger, L.: Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper, Verhand. I. Internat. Math. Kongress (1897), 182-193.Google Scholar

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    Voronoï, G.: On Integral Algebraic Numbers Depending on a Root of an Irreducible Equation of the Third Degree, Master’s dissertation (in Russian), 1894.Google Scholar

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    Voronoï, G.: Sur une propriété du discriminant des fonctions enti`eres, Verhand. III. Internat. Math. Kongress (1905), 186-189.Google Scholar

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    Ward, M.: The characteristic number of a sequences of integers satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), 153-165.Google Scholar

About the article

Received: 2017-05-10

Accepted: 2017-10-18

Published Online: 2018-10-20

Published in Print: 2018-10-25


Citation Information: Mathematica Slovaca, Volume 68, Issue 5, Pages 987–1000, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0161.

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