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Common index divisor of the number fields defined by $x^5+\,ax\,+b$

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  01 December 2022

Anuj Jakhar ,
Sumandeep Kaur  and
Surender Kumar
Anuj Jakhar
Affiliation:
Department of Mathematics, IIT Madras, Chennai 600036, India (anujiisermohali@gmail.com)
Sumandeep Kaur
Affiliation:
Department of Mathematics, Panjab University Chandigarh, Chandigarh 160014, India (sumandhunay@gmail.com)
Surender Kumar
Affiliation:
Department of Mathematics, IIT Bhilai, Raipur 492015, India (syadav2283@gmail.com)
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Abstract

Let $K={\mathbf {Q}}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible polynomial $x^5+ax+b\in {\mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,\,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,\,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.

Keywords

monogenityTheorem of Oreprime ideal factorization

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11R21: Other number fields
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1147 - 1161
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Ahmad, S., Nakahara, T. and Husnine, S. M., Power integral basis for certain pure sextic fields, Int. J. Number Theory 10(8) (2014), 2257–2265.CrossRefGoogle Scholar
Ahmad, S., Nakahara, T. and Hameed, A., On certain pure sextic fields related to a problem of Hasse, Int. J. Algebra Comput. 26(3) (2016), 577–583.CrossRefGoogle Scholar
Cohen, S. D., Movahhedi, A. and Salinier, A., Factorisation over local fields and the irreducibility of generalised difference polynomials, Mathematika 47 (2000), 173–196.CrossRefGoogle Scholar
Fadil, L.E., On non-monogenity of certain number fields defined by trinomials $x^6+ax^3+b$, J. Number Theory 239 (2022), 489–500. doi:10.1016/j.jnt.2021.10.017CrossRefGoogle Scholar
Funakura, T., On integral bases of pure quartic fields, Math. J. Okayama Univ. 26 (1984), 27–41.Google Scholar
Gaál, I., An experiment on the monogenity of a family of trinomials, JP J. Algebra Num. Theory Appl. 51(1) (2021), 97–111.Google Scholar
Gaál, I. and Remete, L., Power integral bases and monogenity of pure fields, J. Num. Theory 173 (2017), 129–146.CrossRefGoogle Scholar
Guárdia, J., Montes, J. and Nart, E., Newton polygons of higher order in algebraic number theory, Trans. Am. Math. Soc. 364(1) (2012), 361–416.CrossRefGoogle Scholar
Jakhar, A. and Kaur, S., A note on non-monogenity of number fields arising from sextic trinomials, Quaestiones Mathematicae (2022), 1–8. doi:10.2989/16073606.2022.2043948CrossRefGoogle Scholar
Jakhar, A. and Kumar, S., On non-monogenic number fields defined by $x^6+ax+b$, Canadian Math. Bull. 65(3) (2022), 788–794.CrossRefGoogle Scholar
Khanduja, S. K., A textbook of algebraic number theory, Unitext series, Volume 135 (Springer, 2022).CrossRefGoogle Scholar
Khanduja, S. K. and Kumar, S., On prolongations of valuations via Newton polygons and liftings of polynomials, J. Pure Appl. Algebra 216 (2012), 2648–2656.CrossRefGoogle Scholar
MacKenzie, R. and Scheuneman, J., A number field without a relative integral basis, Amer. Math. Monthly 78 (1971), 882–823.CrossRefGoogle Scholar
Narkiewicz, W., Elementary and analytic theory of algebraic numbers, 3rd edn., Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2004).CrossRefGoogle Scholar
Neukirch, J., Algebraic number theory, (Berlin-Heidelberg, Springer-Verlag, 1999).CrossRefGoogle Scholar
Smith, H., The monogenity of radical extension, Acta Arith. 198(3) (2021), 313–327.CrossRefGoogle Scholar
Soullami, A. and Sahmoudi, M., On sextic integral bases using relative quadratic extension, Bol. Soc. Paran. Mat. 38(4) (2020), 175–180.Google Scholar