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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, Indiasurenderk@iitbhilai.ac.in
Surender Kumar
Affiliation:
Department of Mathematics, Indian Institute of Technology (IIT) Bhilai, Raipur, Chhattisgarh492015, Indiasurenderk@iitbhilai.ac.in
*
e-mail: anujjakhar@iitbhilai.ac.in
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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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