Distribution of constant terms of irreducible polynomials in ℤp[x] whose degree is a product of two distinct odd primes

Sarah C. Cobb, Michelle L. Knox, Marcos Lopez, Terry McDonald, and Patrick Mitchell
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 1, Pages 195–210
DOI: 10.7546/nntdm.2024.30.1.195-210
Full paper (PDF, 280 Kb)

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Authors and affiliations

Sarah C. Cobb
Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX, 76308 United States

Michelle L. Knox
Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX, 76308 United States

Marcos Lopez
Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX, 76308 United States

Terry McDonald
Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX, 76308 United States

Patrick Mitchell
Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX, 76308 United States

Abstract

We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree q_1q_2 over a finite field, where q_1 and q_2 are distinct odd~primes. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a q_1-residue and/or a q_2-residue in the underlying field. We further show that as k becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal. This paper continues work done in [1].

Keywords

  • Irreducible polynomials
  • Finite fields

2020 Mathematics Subject Classification

  • 11T06
  • 12E05

References

  1. Cobb, S., Knox, M., Lopez, M., McDonald, T., & Mitchell, P. (2019). Distribution of constant terms of polynomials in ℤp[x]. Notes on Number Theory and Discrete Mathematics, 25(4), 72–82.
  2. Křížek, M., Luca, F., & Somer, L. (2001). 17 Lectures on Fermat Numbers. From Number Theory to Geometry. CMS Books in Mathematics, Springer-Verlag, New York.
  3. Lidl, R., & Niederreiter, H. (1994). Introduction to Finite Fields and Their Applications (Revised ed.). Cambridge UP, Cambridge.
  4. Omidi Koma, B., Panario, D., & Wang, Q. (2010). The number of irreducible polynomials of degree n over \mathbb{F}_q with given trace and constant terms. Discrete Mathematics, 310, 1282–1292.
  5. Yucas, J. L. (2006). Irreducible polynomials over finite fields with prescribed trace/prescribed constant term. Finite Fields and Their Applications 12, 211–221.

Manuscript history

  • Received: 27 September 2023
  • Accepted: 6 March 2024
  • Online First: 28 March 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Cobb, S. C., Knox, M. L., Lopez, M., McDonald, T., & Mitchell, P. (2024). Distribution of constant terms of irreducible polynomials in ℤp[x] whose degree is a product of two distinct odd primes. Notes on Number Theory and Discrete Mathematics, 30(1), 195-210, DOI: 10.7546/nntdm.2024.30.1.195-210.

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