Home Constructing infinite families of number fields with given indices from quintinomials
Article
Licensed
Unlicensed Requires Authentication

Constructing infinite families of number fields with given indices from quintinomials

  • Lhoussain El Fadil and Omar Kchit EMAIL logo
Published/Copyright: February 25, 2025
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 75 Issue 1

Abstract

In this paper, for any rational prime p and for a fixed positive integer νp, we provide infinite families of number fields defined by quintinomials satisfying νp(i(K)) = νp. We illustrate our results by some computational examples.

Keywords: Theorem of Dedekind; Theorem of Ore; prime ideal factorization; Newton polygon; index of a number field; monogenic
MSC 2010: Primary 11R04; 11Y40; 11R21

Acknowledgement

The authors are deeply grateful to the anonymous referee whose valuable comments and suggestions have tremendously improved the quality of this paper. They also extend their sincere thanks to Professor István Gaál for his advice and encouragement. The first author is very grateful to Professor Enric Nart who introduced him to Newton’s polygon techniques.

  1. (Communicated by István Gaál)

References

[1] Cohen, H.: A Course in Computational Algebraic Number Theory, Gtm 138, Springer-Verlag Berlin, Heidelberg, 1993.10.1007/978-3-662-02945-9Search in Google Scholar

[2] Davis, C. T.—Spearman, B. K.: The index of a quartic field defined by a trinomial x4 + ax + b, J. Algebra Appl. 17(10) (2018), 185–197.10.1142/S0219498818501979Search in Google Scholar

[3] Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Göttingen Abhandlungen 23 (1878), 1–23.Search in Google Scholar

[4] EL Fadil, L.: On the index divisors and monogenity of number fields defined x5 + ax3 + b, Quaest. Math. 46(11) (2023), 2355–2365.10.2989/16073606.2022.2156000Search in Google Scholar

[5] EL Fadil, L.: On indices of septic number fields defined by trinomials x7 + ax + b, Mathematics 11(21) (2023), Art. No. 4441.10.3390/math11214441Search in Google Scholar

[6] EL Fadil, L.—Gaál, I.: On non-monogenity of certain number fields defined by trinomials x4 + ax2 + b, (2022), https://doi.org/10.48550/arXiv.2204.03226.10.1515/ms-2023-0063Search in Google Scholar

[7] EL Fadil, L.—Kchit, O.: On index divisors and monogenity of certain sextic number fields defined by x6 + ax5 + b, Vietnam J. Math. (2024), https://doi.org/10.1007/s10013-023-00679-3.10.1007/s10013-023-00679-3Search in Google Scholar

[8] EL Fadil, L.—Kchit, O.: On index divisors and monogenity of certain septic number fields defined by x7 + ax3 + b, Comm. Algebra 51(6) (2023), 2349–2363.10.1080/00927872.2022.2159035Search in Google Scholar

[9] EL Fadil, L.—Kchit, O.: On index divisors and monogenity of certain number fields defined by x12 + axm + b, Ramanujan J. 63 (2024), 451–482.10.1007/s11139-023-00768-4Search in Google Scholar

[10] EL Fadil, L.—Montes, J.—Nart, E.: Newton polygons and p-integral bases of quartic number fields, J. Algebra Appl. 11(4) (2012), Art. ID 1250073.10.1142/S0219498812500739Search in Google Scholar

[11] Engstrom, H. T.: On the common index divisor of an algebraic number field, Trans. Amer. Math. Soc. 32 (1930), 223–237.10.1090/S0002-9947-1930-1501535-0Search in Google Scholar

[12] Funakura, T.: On integral bases of pure quartic fields, Math. J. Okayama Univ. 26 (1984), 27–41.Search in Google Scholar

[13] Gaál, I.: Diophantine Equations and Power Integral Bases, Theory and Algorithm, 2nd ed., Boston, Birkhäuser, 2019.10.1007/978-3-030-23865-0Search in Google Scholar

[14] Gaál, I.—Pethö, A.—Pohst, M.: On the indices of biquadratic number fields having Galois group V4, Arch. Math. (Basel) 57 (1991), 357–361.10.1007/BF01198960Search in Google Scholar

[15] Gaál, I.—Pethö, A.—Pohst, M.: On the resolution of index form equations in quartic number fields, J. Symbolic Comput. 16(6) (1993), 563–584.10.1006/jsco.1993.1064Search in Google Scholar

[16] Guàrdia, J.—Montes J.—Nart, E.: Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012), 361–416.10.1090/S0002-9947-2011-05442-5Search in Google Scholar

[17] Guàrdia, J.—Nart, E.: Genetics of polynomials over local fields, Contemp. Math. 637 (2015), 207–241.10.1090/conm/637/12767Search in Google Scholar

[18] Hensel, K.: Theorie der Algebraischen Zahlen, Teubner Verlag, Leipzig, Berlin, 1908.Search in Google Scholar

[19] Kchit, O.: On the index divisors and monogenity of certain nonic number fields, Rocky Mountain J. Math. (2023), in press.Search in Google Scholar

[20] Kchit, O.: On the index of certain nonic number fields defined by x9 + ax5 + b, P-Adic Numbers Ultrametric Anal. Appl. 16 (2024), 95–112.10.1134/S2070046624020018Search in Google Scholar

[21] Kchit, O.: On index divisors and non-monogenity of certain quintic number fields defined by x5 + axm + bx + c, Comm. Algebra 51(8) (2023), 3172–3181.10.1080/00927872.2023.2179633Search in Google Scholar

[22] Nakahara, T.: On the indices and integral bases of non-cyclic but abelian biquadratic fields, Arch. Math. (Basel) 41(6) (1983), 504–508.10.1007/BF01198579Search in Google Scholar

[23] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, Springer Verlag, 3rd ed., 2004.10.1007/978-3-662-07001-7Search in Google Scholar

[24] Nart, E.: On the index of a number field, Trans. Amer. Math. Soc. 289 (1985), 171–183.10.1090/S0002-9947-1985-0779058-2Search in Google Scholar

[25] Neukirch, J.: Algebraic Number Theory, Springer-Verlag, Berlin, 1999.10.1007/978-3-662-03983-0Search in Google Scholar

[26] Ore, O.: Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann. 99 (1928), 84–117.10.1007/BF01459087Search in Google Scholar

[27] Pethö, A.—Pohst, M.: On the indices of multiquadratic number fields, Acta Arith. 153 (2012), 393–414.10.4064/aa153-4-4Search in Google Scholar

[28] Śliwa, J.: On the nonessential discriminant divisor of an algebraic number field, Acta Arith. 42 (1982), 57–72.10.4064/aa-42-1-57-72Search in Google Scholar

[29] Spearman, B. K.—Williams, K. S.: The index of a cyclic quartic field, Monatsh. Math. 140 (2003), 19–70.10.1007/s00605-002-0547-3Search in Google Scholar
Received: 2024-05-22
Accepted: 2024-07-13
Published Online: 2025-02-25
Published in Print: 2025-02-25

© 2025 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Generalized Sasaki mappings in d0-Algebras
  2. On a theorem of Nathanson on Diophantine approximation
  3. Constructing infinite families of number fields with given indices from quintinomials
  4. Partitions into two Lehmer numbers in ℤq
  5. Fundamental systems of solutions of some linear differential equations of higher order
  6. On k-Circulant matrices involving the Lucas numbers of even index
  7. Explicit formulae for the Drazin inverse of the sum of two matrices
  8. On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
  9. Novel generalized tempered fractional integral inequalities for convexity property and applications
  10. Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
  11. The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
  12. Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
  13. Improvements on the Leighton oscillation theorem for second-order dynamic equations
  14. Topogenous orders on forms
  15. Comparison of topologies on fundamental groups with subgroup topology viewpoint
  16. An elementary proof of the generalized Itô formula
  17. Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
  18. Advancing reliability and medical data analysis through novel statistical distribution exploration
  19. Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
https://doi.org/10.1515/ms-2025-0003
Keywords for this article
Theorem of Dedekind; Theorem of Ore; prime ideal factorization; Newton polygon; index of a number field; monogenic

Articles in the same Issue

  1. Generalized Sasaki mappings in d0-Algebras
  2. On a theorem of Nathanson on Diophantine approximation
  3. Constructing infinite families of number fields with given indices from quintinomials
  4. Partitions into two Lehmer numbers in ℤq
  5. Fundamental systems of solutions of some linear differential equations of higher order
  6. On k-Circulant matrices involving the Lucas numbers of even index
  7. Explicit formulae for the Drazin inverse of the sum of two matrices
  8. On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
  9. Novel generalized tempered fractional integral inequalities for convexity property and applications
  10. Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
  11. The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
  12. Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
  13. Improvements on the Leighton oscillation theorem for second-order dynamic equations
  14. Topogenous orders on forms
  15. Comparison of topologies on fundamental groups with subgroup topology viewpoint
  16. An elementary proof of the generalized Itô formula
  17. Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
  18. Advancing reliability and medical data analysis through novel statistical distribution exploration
  19. Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
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 25.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0003/html
Scroll to top button