Skip to main content
SpringerLink
Log in

On Index Divisors and Monogenity of Certain Sextic Number Fields Defined by \(x^6+ax^5+b\)

  • Original Article
  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz (2004) for any sextic number field K generated by a root of a monic irreducible trinomial \(F(x)=x^6+ax^5+b\in \mathbb {Z}[x]\). Namely, we calculate the index of the field K. In particular, if \(i(K)\ne 1\), then K is not mongenic. Finally, we illustrate our results by some computational examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alaca, S., Williams, K.S.: Introductory Algebraic Number Theory. Cambridge University Press, New York (2004)

    Google Scholar 

  2. Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin, Heidelberg (1993)

  3. Davis, C.T., Spearman, B.K.: The index of a quartic field defined by a trinomial \(X^4+aX+b\). J. Algebra Appl. 17, 1850197 (2018)

    Article  MathSciNet  Google Scholar 

  4. Deajim, A., El Fadil, L.: On the integral closedness of \(R[\alpha ]\). Math. Rep. 24, 571–581 (2022)

    MathSciNet  Google Scholar 

  5. Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen. Göttingen Abhandlungen 23, 1–23 (1878)

    Google Scholar 

  6. El Fadil, L.: On index and monogenity of certain number fields defined by trinomials. Math. Slovaca 73, 861–870 (2023)

    Article  MathSciNet  Google Scholar 

  7. El Fadil, L.: On common index divisors and monogenity of certain number fields defined by \(x^5+ax^2+b\). Commun. Algebra 50, 3102–3113 (2022)

    Article  Google Scholar 

  8. El Fadil, L.: On non monogenity of certain number fields defined by a trinomial \(x^6+ax^3+b\). J. Number Theory 239, 489–500 (2022)

    Article  MathSciNet  Google Scholar 

  9. El Fadil, L., Gaál, I.: On non-monogenity of certain number fields defined by trinomials \(x^4+ax^2+b\). arXiv:2204.03226 (2022)

  10. El Fadil, L., Kchit, O.: On index divisors and monogenity of certain septic number fields defined by \(x^7+ax^3+b\). Commun. Algebra 53, 2349–2363 (2023)

    Article  Google Scholar 

  11. El Fadil, L., Montes, J., Nart, E.: Newton polygons and \(p\)-integral bases of quartic number fields. J. Algebra Appl. 11, 1250073 (2012)

    Article  MathSciNet  Google Scholar 

  12. Engler, A.J., Prestel, A.: Valued Fields. Springer, Berlin, Heidelberg (2005)

    Google Scholar 

  13. Engstrom, H.T.: On the common index divisor of an algebraic number field. Trans. Amer. Math. Soc. 32, 223–237 (1930)

    Article  MathSciNet  Google Scholar 

  14. Gaál, I.: An experiment on the monogenity of a family of trinomials. JP J. Algebra Number Theory Appl. 51, 97–111 (2021)

    Google Scholar 

  15. Gaál, I., Pethö, A., Pohst, M.: On the indices of biquadratic number fields having Galois group \(V_4\). Arch. Math. 57, 357–361 (1991)

    Article  MathSciNet  Google Scholar 

  16. Guàrdia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory. Trans. Amer. Math. Soc. 364, 361–416 (2012)

    Article  MathSciNet  Google Scholar 

  17. Hensel, K.: Arithmetische Untersuchungen über die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung. J. Reine Angew. Math. 113, 128–160 (1894)

    MathSciNet  Google Scholar 

  18. Ibarra, R., Lembeck, H., Ozaslan, M., Smith, H., Stange, K.E.: Monogenic fields arising from trinomials. Involve J. Math. 15, 299–317 (2022)

    Article  MathSciNet  Google Scholar 

  19. Montes, J., Nart, E.: On a theorem of Ore. J. Algebra 146, 318–334 (1992)

    Article  MathSciNet  Google Scholar 

  20. Nakahara, T.: On the indices and integral bases of non-cyclic but abelian biquadratic fields. Arch. Math. 41, 504–508 (1983)

    Article  MathSciNet  Google Scholar 

  21. Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer, Berlin, Heidelberg (2004)

    Book  Google Scholar 

  22. Neukirch, J.: Algebraic Number Theory. Springer, Berlin (1999)

    Book  Google Scholar 

  23. Ore, Ö.: Newtonsche Polygone in der Theorie der algebraischen Körper. Math. Ann. 99, 84–117 (1928)

    Article  MathSciNet  Google Scholar 

  24. von Żyliński, E.: Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer Körper. Math. Ann. 73, 273–274 (1913)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Faculty of Sciences Dhar El Mahraz, Sidi Mohamed ben Abdellah University, P.O. Box 1796, Atlas-Fes, Morocco

    Lhoussain El Fadil & Omar Kchit

Authors
  1. Lhoussain El Fadil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omar Kchit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lhoussain El Fadil.

Ethics declarations

Competing interests

There are non-financial competing interests to report.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Fadil, L., Kchit, O. On Index Divisors and Monogenity of Certain Sextic Number Fields Defined by \(x^6+ax^5+b\). Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-023-00679-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10013-023-00679-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation