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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

Information

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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