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On nonmonogenic number fields defined by $x^6+ax+b$

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  15 September 2021

Anuj Jakhar  and
Surender Kumar
Anuj Jakhar*
Affiliation:
Department of Mathematics, Indian Institute of Technology (IIT) Bhilai, Raipur, Chhattisgarh492015, [email protected]
Surender Kumar
Affiliation:
Department of Mathematics, Indian Institute of Technology (IIT) Bhilai, Raipur, Chhattisgarh492015, [email protected]
*
e-mail: [email protected]
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Abstract

Let q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.

Keywords

MonogenitynonmonogenityNewton polygonpower basis

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 788 - 794
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Copyright
© Canadian Mathematical Society 2021

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References

Ahmad, S., Nakahara, T., and Hameed, A., On certain pure sextic fields related to a problem of Hasse . Int. J. Algebra Comput. 26(2016), no. 3, 577–583.10.1142/S0218196716500259CrossRefGoogle Scholar
Bilu, Y., Gaál, I., and Györy, K., Index form equations in sextic fields: a hard computation . Acta Arith. 115(2004), no. 1, 85–96.10.4064/aa115-1-7CrossRefGoogle Scholar
Dedekind, R., Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen . Göttingen Abh. 23(1878), 1–23.Google Scholar
Fadil, L. E., On integral bases and monogenity of pure sextic number fields with non-squarefree coefficients . J. Number Theory 228(2021), 375–389.CrossRefGoogle Scholar
Gaál, I., Power integral bases in algebraic number fields . Ann. Univ. Sci. Budapest. Sect. Comput. 18(1999), 61—87.Google Scholar
Gaál, I., Diophantine equations and power integral bases, theory and algorithm. 2nd ed., Birkhäuser, Boston, 2019.CrossRefGoogle Scholar
Gaál, I., Olajos, P., and Pohst, M., Power integral bases in orders of composite fields . Exp. Math. 11(2002), no. 1, 87–90.10.1080/10586458.2002.10504471CrossRefGoogle Scholar
Gaál, I. and Remete, L., Power integral bases and monogenity of pure fields . J. Number Theory 173(2017), 129–146.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
Pethö, A. and Pohst, M., On the indices of multiquadratic number fields . Acta Arith. 153(2012), no. 4, 393–414.10.4064/aa153-4-4CrossRefGoogle Scholar
Yakkou, H. B. and Fadil, L. E., On monogenty of certain pure number fields defined by ${x}^{p^r}-m$ . Int. J. Number Theory (2021). https://doi.org/10.1142/S1793042121500858 Google Scholar