Home Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three
Article
Licensed
Unlicensed Requires Authentication

Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three

  • Jiří Klaška EMAIL logo and Ladislav Skula
Published/Copyright: November 3, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 4

Abstract

In this paper we extend our recent results concerning the validity of the law of inertia for the factorization of cubic polynomials over the Galois field Fp, p being a prime. As the main result, the following theorem will be proved: Let D∈Z and let CD be the set of all cubic polynomials x3+ax2+bx+c∈Z[x] with a discriminant equal to D. If D is square-free and 3∤h(−3D) where h(−3D) is the class number of Q(−3D), then all cubic polynomials in CD have the same type of factorization over any Galois field Fp where p is a prime, p > 3.

MSC 2010: Primary 11T06; 11D25; 11D05
keywords: cubic polynomial; type of factorization; discriminant

(Communicated by Stanislav Jakubec)

Acknowledgement

The authors wish to express their appreciation to the anonymous referee for the careful reading and the helpful comments and suggestions that improved the quality of this paper.

References

[1] Klaška, J.—Skula, L.: Mordell’s equation and the Tribonacci family, Fibonacci Quart. 49 (2011), 310–319.10.1080/00150517.2011.12428030Search in Google Scholar

[2] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials – the real case, Util. Math. (To appear).Search in Google Scholar

[3] Klaška, J.—Skula, L.: Law of inertia for the factorization of cubic polynomials – the imaginary case, Util. Math. (To appear).Search in Google Scholar

[4] London, J.—Finkelstein, M.: On Mordell’s Equation y2 –k = x3, Ohio Bowling Green State University, Bowling Green, 1973.Search in Google Scholar

[5] Stickelberger, L.: Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper. In: Verhand. I. Internat. Math. Kongress Zurich, 1897, Leipzig, 1898, pp. 182–193.Search in Google Scholar

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

[7] Voronoï, G.: Sur une propriété du discriminant des fonctions entières. In: Verhand. III. Internat. Math. Kongress Heidelberg, Leipzig, 1905, pp. 186–189.Search in Google Scholar

[8] Ward, M.: The characteristic number of a sequences of integers satisfying a linear recursion relation, Trans. Amer. Math. Soc. 33 (1931), 153–165.10.1090/S0002-9947-1931-1501582-XSearch in Google Scholar

Received: 2013-10-15
Accepted: 2014-3-25
Published Online: 2016-11-3
Published in Print: 2016-8-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. Effect algebras with a full set of states
  3. Research Article
  4. An approach to orthomodular lattices via lattices with an antitone involution
  5. Research Article
  6. On properties of the free BCI-algebra with one generator
  7. Research Article
  8. The generalized order-k Narayana's cows numbers
  9. Research Article
  10. The irrationality of infinite series of a special kind
  11. Research Article
  12. On skew-commuting mappings in semiprime rings
  13. Research Article
  14. On the subordination and superordination of strongly starlike functions
  15. Research Article
  16. On a subclass of analytic functions involving Sălăgean integral operator
  17. Research Article
  18. Uniqueness of meromorphic functions sharing a value or small function
  19. Research Article
  20. A new inclusion for Bavrin's families of holomorphic functions in n-circular domains
  21. Research Article
  22. Oscillation of nonlinear fourth order mixed neutral differential equations
  23. Research Article
  24. Oscillation results for third order nonlinear mixed neutral differential equations
  25. Research Article
  26. Homoclinic solutions for second order Hamiltonian systems with general potentials
  27. Research Article
  28. A class of fractional impulsive functional differential equations with nonlocal conditions
  29. Research Article
  30. The weighted reverse poincaré type inequality for the difference of two parabolic subsolutions
  31. Research Article
  32. On the well posed solutions for nonlinear second order neutral difference equations
  33. Research Article
  34. Some approximation results for operators of Szász-Mirakjan-Durrmeyer type
  35. Research Article
  36. On the geometry of conditional expectations treated as projections on the L2-space
  37. Research Article
  38. Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators
  39. Research Article
  40. A necessary condition for the Smith equivalence of G-modules and its sufficiency
  41. Research Article
  42. On the internal approach to differential equations. 1. The involutiveness and standard basis
  43. Research Article
  44. Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three
https://doi.org/10.1515/ms-2015-0199
Keywords for this article
cubic polynomial; type of factorization; discriminant

Articles in the same Issue

  1. Research Article
  2. Effect algebras with a full set of states
  3. Research Article
  4. An approach to orthomodular lattices via lattices with an antitone involution
  5. Research Article
  6. On properties of the free BCI-algebra with one generator
  7. Research Article
  8. The generalized order-k Narayana's cows numbers
  9. Research Article
  10. The irrationality of infinite series of a special kind
  11. Research Article
  12. On skew-commuting mappings in semiprime rings
  13. Research Article
  14. On the subordination and superordination of strongly starlike functions
  15. Research Article
  16. On a subclass of analytic functions involving Sălăgean integral operator
  17. Research Article
  18. Uniqueness of meromorphic functions sharing a value or small function
  19. Research Article
  20. A new inclusion for Bavrin's families of holomorphic functions in n-circular domains
  21. Research Article
  22. Oscillation of nonlinear fourth order mixed neutral differential equations
  23. Research Article
  24. Oscillation results for third order nonlinear mixed neutral differential equations
  25. Research Article
  26. Homoclinic solutions for second order Hamiltonian systems with general potentials
  27. Research Article
  28. A class of fractional impulsive functional differential equations with nonlocal conditions
  29. Research Article
  30. The weighted reverse poincaré type inequality for the difference of two parabolic subsolutions
  31. Research Article
  32. On the well posed solutions for nonlinear second order neutral difference equations
  33. Research Article
  34. Some approximation results for operators of Szász-Mirakjan-Durrmeyer type
  35. Research Article
  36. On the geometry of conditional expectations treated as projections on the L2-space
  37. Research Article
  38. Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators
  39. Research Article
  40. A necessary condition for the Smith equivalence of G-modules and its sufficiency
  41. Research Article
  42. On the internal approach to differential equations. 1. The involutiveness and standard basis
  43. Research Article
  44. Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0199/html
Scroll to top button