Hostname: page-component-77f85d65b8-g4pgd Total loading time: 0 Render date: 2026-03-27T16:14:01.897Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS

Part of: Computational number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  01 April 2024

ANUJ JAKHAR Open the ORCID record for ANUJ JAKHAR [Opens in a new window]
ANUJ JAKHAR*
Affiliation:
Department of Mathematics, Indian Institute of Technology (IIT) Madras, Chennai, India e-mail: anujjakhar@iitm.ac.in
*
e-mail: anujiisermohali@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$, where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$, $1\leq m<n$. We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$. We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$. As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$, the symmetric group on n letters.

Keywords

rings of algebraic integersindex of an algebraic integerpower basis

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11R29: Class numbers, class groups, discriminants 11Y40: Algebraic number theory computations

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 460 - 467
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The author is thankful to IIT Madras for NFIG grant RF/22-23/1035/MA/NFIG/009034.

Dedicated to Professor Sudesh Kaur Khanduja

References

Bishnoi, A. and Khanduja, S. K., ‘A class of trinomials with Galois group ${S}_n$ ’, Algebra Colloq. 19(1) (2012), 905–911.CrossRefGoogle Scholar
Cohen, H., A Course in Computational Algebraic Number Theory (Springer-Verlag, Berlin–Heidelberg, 1993).CrossRefGoogle Scholar
Dedekind, R., ‘Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen’, Götttingen Abh. 23 (1878), 1–23.Google Scholar
Filaseta, M. and Moy, R., ‘On the Galois group over $\mathbb{Q}$ of a truncated binomial expansion’, Colloq. Math. 154 (2018), 295–308.CrossRefGoogle Scholar
Gaál, I., Diophantine Equations and Power Integral Bases: Theory and Algorithms, 2nd edn (Birkhäuser/Springer, Cham, 2019).CrossRefGoogle Scholar
Hasse, H., Zahlentheorie (Akademie-Verlag, Berlin, 1963).CrossRefGoogle Scholar
Jakhar, A., Khanduja, S. K. and Sangwan, N., ‘On prime divisors of the index of an algebraic integer’, J. Number Theory 166 (2016), 47–61.CrossRefGoogle Scholar
Jakhar, A., Khanduja, S. K. and Sangwan, N., ‘Characterisation of primes dividing the index of a trinomial’, Int. J. Number Theory 13(10) (2017), 2505–2514.CrossRefGoogle Scholar
Jones, L., ‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc. 100 (2019), 239–244.CrossRefGoogle Scholar
Jones, L., ‘On necessary and sufficient conditions for the monogenity of a certain class of polynomials’, Math. Slovaca 72(3) (2022), 591–600.CrossRefGoogle Scholar
Jones, L., ‘Infinite families of monogenic quadrinomials, quintinomials and sextinomials’, Colloq. Math. 169 (2022), 1–10.CrossRefGoogle Scholar
Kaur, S. and Kumar, S., ‘On a conjecture of Lenny Jones about certain monogenic polynomials’, Bull. Aust. Math. Soc., to appear. Published online (21 November 2023).CrossRefGoogle Scholar