Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-29T00:20:55.265Z Has data issue: false hasContentIssue false
  • English
  • Français

NON-PÓLYA FIELDS WITH LARGE PÓLYA GROUPS ARISING FROM LEHMER QUINTICS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  11 March 2024

NIMISH KUMAR MAHAPATRA Open the ORCID record for NIMISH KUMAR MAHAPATRA [Opens in a new window]  and
PREM PRAKASH PANDEY Open the ORCID record for PREM PRAKASH PANDEY [Opens in a new window]
NIMISH KUMAR MAHAPATRA*
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research, Berhampur, India
PREM PRAKASH PANDEY
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research, Berhampur, India e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a new family of quintic non-Pólya fields with large Pólya groups. We show that the Pólya number of such a field never exceeds five times the size of its Pólya group. Finally, we show that these non-Pólya fields are nonmonogenic of field index one.

Keywords

Pólya fieldPólya groupHilbert class fieldgenus field

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R04: Algebraic numbers; rings of algebraic integers 11R20: Other abelian and metabelian extensions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 468 - 479
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

N.K.M. would like to acknowledge financial support from the University Grants Commission (UGC), Government of India.

References

Cahen, P. J. and Chabert, J. L., Integer-Valued Polynomials, Mathematical Surveys and Monographs, 48 (American Mathematical Society, Providence, RI, 1997).Google Scholar
Chabert, J. L., ‘Factorial groups and Pólya groups in Galoisian extensions of $\mathbb{Q}$ ’, in: Commutative Ring Theory and Applications, Lecture Notes in Pure and Applied Mathematics, 231 (eds. Fontana, M., Kabbaj, S.-E. and Wiegand, S.) (Dekker, New York, 2003), 77–86.Google Scholar
Chattopadhyay, J. and Saikia, A., ‘Totally real bi-quadratic fields with large Pólya groups’, Res. Number Theory 8 (2022), Article no. 30.CrossRefGoogle Scholar
Erdős, P., ‘Arithmetical properties of polynomials’, J. Lond. Math. Soc. (2) 28 (1953), 416–425.CrossRefGoogle Scholar
Gillibert, J. and Levin, A., ‘A geometric approach to large class groups: A survey’, in: Class Groups of Number Fields and Related Topics (eds. Chakraborty, K., Hoque, A. and Pandey, P.) (Springer, Singapore, 2020).Google Scholar
Golod, E. S. and Shafarevich, I. R., ‘On the class field tower’, Izv. Akad. Nauk 28 (1964), 261–272.Google Scholar
Gras, M.-N., ‘Non monogénéité de l’anneau des entires des extensions cycliques de $\mathbb{Q}$ de degré premier $l\ge 5$ ’, J. Number Theory 23 (1986), 347–353.CrossRefGoogle Scholar
Halberstam, H., ‘On the distribution of additive number-theoretic functions. III’, J. Lond. Math. Soc. (2) 31 (1956), 14–27.CrossRefGoogle Scholar
Heidaryan, B. and Rajaei, A., ‘Biquadratic Pólya fields with only one quadratic Pólya subfield’, J. Number Theory 143 (2014), 279–285.CrossRefGoogle Scholar
Heidaryan, B. and Rajaei, A., ‘Some non-Pólya biquadratic fields with low ramification’, Rev. Mat. Iberoam. 33 (2017), 1037–1044.CrossRefGoogle Scholar
Helfgott, H., ‘Power-free values, large deviations and integer points on irrational curves’, J. Théor. Nombres Bordeaux 19 (2007), 433–472.CrossRefGoogle Scholar
Helfgott, H., ‘Power-free values repulsion between points, differing beliefs and the existence of error’, in: Anatomy of Integers, CRM Proceedings and Lecture Notes, 46 (eds. J.-M. De Koninck, A. Granville and F. Luca) (American Mathematical Society, Providence, RI, 2008), 81–88.CrossRefGoogle Scholar
Hilbert, D., ‘Die Theorie der algebraischen Zahlkörper’, Jahresber. Dtsch. Math.-Ver. 4 (1894–1895), 175–546.Google Scholar
Ishida, M., The Genus Fields of Algebraic Number Fields (Springer, Cham, 1976).CrossRefGoogle Scholar
Jeannin, S., ‘Nombre de classes et unités des corps de nombres cycliques quintiques d’E. Lehmer’, J. Théor. Nombres Bordeaux 8 (1996), 75–92.CrossRefGoogle Scholar
Lehmer, E., ‘Connection between Gaussian periods and cyclic units’, Math. Comp. 50 (1988), 535–541.CrossRefGoogle Scholar
Leriche, A., ‘Cubic, quartic and sextic Pólya fields’, J. Number Theory 133 (2013), 59–71.CrossRefGoogle Scholar
Leriche, A., ‘About the embedding of a number field in a Pólya field’, J. Number Theory 145 (2014), 210–229.CrossRefGoogle Scholar
Maarefparvar, A., ‘Pólya group in some real biquadratic fields’, J. Number Theory 228 (2021), 1–7.CrossRefGoogle Scholar
Masser, D. W., ‘Polynomial bounds for Diophantine equations’, Amer. Math. Monthly 93 (1986), 486–488.CrossRefGoogle Scholar
Reuss, T., ‘Power-free values of polynomials’, Bull. Lond. Math. Soc. 47 (2015), 270–284.CrossRefGoogle Scholar
Schoof, R. and Washington, L. C., ‘Quintic polynomials and real cyclotomic fields with large class numbers’, Math. Comp. 50 (1988), 543–556.Google Scholar
Tougma, C. W.-W., ‘Some questions on biquadratic Pólya fields’, J. Number Theory 229 (2021), 386–398.CrossRefGoogle Scholar
von Zylinski, E., ‘Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer Körper’, Math. Ann. 73(2) (1913), 273–274.CrossRefGoogle Scholar
Wolske, Z., Number Fields with Large Minimal Index. Thesis, University of Toronto, 2018.Google Scholar
Zantema, H., ‘Integer valued polynomials over a number field’, Manuscripta Math. 40 (1982), 155–203.CrossRefGoogle Scholar