**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)**

## Details

### 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

3410 Taft Blvd, Wichita Falls, TX, 76308 United States

Patrick Mitchell

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 over a finite field, where and 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 -residue and/or a -residue in the underlying field. We further show that as 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

- Cobb, S., Knox, M., Lopez, M., McDonald, T., & Mitchell, P. (2019). Distribution of constant terms of polynomials in ℤ
_{p}[*x*].*Notes on Number Theory and Discrete Mathematics*, 25(4), 72–82. - 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. - Lidl, R., & Niederreiter, H. (1994).
*Introduction to Finite Fields and Their Applications*(Revised ed.). Cambridge UP, Cambridge. - Omidi Koma, B., Panario, D., & Wang, Q. (2010). The number of irreducible polynomials of degree over with given trace and constant terms.
*Discrete Mathematics*, 310, 1282–1292. - 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).

## Related papers

- Cobb, S., Knox, M., Lopez, M., McDonald, T., & Mitchell, P. (2019). Distribution of constant terms of polynomials in ℤ
_{p}[*x*].*Notes on Number Theory and Discrete Mathematics*, 25(4), 72–82.

## Cite this paper

Cobb, S. C., Knox, M. L., Lopez, M., McDonald, T., & Mitchell, P. (2024). Distribution of constant terms of irreducible polynomials in ℤ_{p}[*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.