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Arithmetic progression in a finite field with prescribed norms

  • Kaustav Chatterjee , Hariom Sharma , Aastha Shukla and Shailesh Kumar Tiwari ORCID logo EMAIL logo
Published/Copyright: March 26, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 1

Abstract

Given a prime power q and a positive integer n, let 𝔽 q n represent a finite extension of degree n of the finite field 𝔽 q . In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n ≥ 6 , q = 3 k , m = 2 we establish that there are only 10 possible exceptions.

Keywords: Finite field; primitive element; normal element; characters; norm
MSC 2020: 12E20; 11T23

Communicated by Freydoon Shahidi

Funding statement: First author is supported by the National Board for Higher Mathematics (IN), Ref No. 0203/6/2020-R&D-II/7387.

Acknowledgements

We sincerely appreciate and acknowledge the reviewers for their helpful comments and suggestions. All authors are equally contributed.

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Received: 2024-01-12
Revised: 2024-02-22
Published Online: 2024-03-26
Published in Print: 2025-01-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Some properties of extended frame measure
  3. Orthogonal separation of variables for spaces of constant curvature
  4. Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications
  5. Uniform bounds for Kloosterman sums of half-integral weight with applications
  6. q-supercongruences from Watson's 8φ7 transformation
  7. Submodules of normalisers in groupoid C*-algebras and discrete group coactions
  8. Explicit bounds for the solutions of superelliptic equations over number fields
  9. The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth
  10. A Kollár-type vanishing theorem for k-positive vector bundles
  11. New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators
  12. On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting
  13. A p-adic analog of Hasse--Davenport product relation involving ϵ-factors
  14. On arithmetic quotients of the group SL2 over a quaternion division k-algebra
  15. Euler’s integral, multiple cosine function and zeta values
  16. Paley inequality for the Weyl transform and its applications
  17. Homogeneous ACM and Ulrich bundles on rational homogeneous spaces
  18. Arithmetic progression in a finite field with prescribed norms
https://doi.org/10.1515/forum-2024-0026
Keywords for this article
Finite field; primitive element; normal element; characters; norm

Articles in the same Issue

  1. Frontmatter
  2. Some properties of extended frame measure
  3. Orthogonal separation of variables for spaces of constant curvature
  4. Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications
  5. Uniform bounds for Kloosterman sums of half-integral weight with applications
  6. q-supercongruences from Watson's 8φ7 transformation
  7. Submodules of normalisers in groupoid C*-algebras and discrete group coactions
  8. Explicit bounds for the solutions of superelliptic equations over number fields
  9. The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth
  10. A Kollár-type vanishing theorem for k-positive vector bundles
  11. New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators
  12. On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting
  13. A p-adic analog of Hasse--Davenport product relation involving ϵ-factors
  14. On arithmetic quotients of the group SL2 over a quaternion division k-algebra
  15. Euler’s integral, multiple cosine function and zeta values
  16. Paley inequality for the Weyl transform and its applications
  17. Homogeneous ACM and Ulrich bundles on rational homogeneous spaces
  18. Arithmetic progression in a finite field with prescribed norms
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