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 25, 2019, Number 4, Pages 72–82

DOI: 10.7546/nntdm.2019.25.4.72-82

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## Details

### Authors and affiliations

Sarah C. Cobb

*Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX 76308 USA
*

Michelle L. Knox

*Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX 76308 USA
*

Marcos Lopez

*Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX 76308 USA
*

Terry McDonald

*Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX 76308 USA
*

Patrick Mitchell

*Department of Mathematics, Midwestern State University
3410 Taft Blvd, Wichita Falls, TX 76308 USA
*

### Abstract

We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree *q ^{k}* over a finite field. 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 in the underlying field. We further show that as

*k*becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal.

### Keywords

- Irreducible polynomials
- Finite fields

### 2010 Mathematics Subject Classification

- 11T06
- 12E05

### References

- Krizek, M., Luca, F., & Somer, L. (2001). 17 Lectures on Fermat Numbers, CMS Books in Mathematics, Springer-Verlag, New York.
- Lidl, R., & Niederreiter, H. (1994). Introduction to Finite Fields and Their Application, Revised edition, Cambridge University Press, Cambridge.
- Niven, I., Zuckerman, H., & Montgomery, H. (1991). An Introduction to the Theory of Numbers, 5th edition, Wiley and Sons, Inc., New York.
- Rubinstein, M., & Sarnak, P. (1994). Chebyshev’s Bias, Experimental Mathematics, 3, 173–197.
- Yucas, J. L. (2006). Irreducible polynomials over finite fields with prescribed trace/prescribed constant term, Finite Fields and Their Appl, 12, 211–221.

## Related papers

## Cite this paper

APACobb, S. C., Knox, M. L., Lopez, M., McDonald, T. & Mitchell, P. (2019). Distribution of constant terms of irreducible polynomials in ℤ_{p}[*x*]. Notes on Number Theory and Discrete Mathematics, 25(4), 72-82, doi: 10.7546/nntdm.2019.25.4.72-82.

Cobb, Sarah C., Michelle L. Knox, Marcos Lopez, Terry McDonald and Patrick Mitchell. “Distribution of Constant Terms of Irreducible Polynomials in ℤ_{p}[*x*].” Notes on Number Theory and Discrete Mathematics 25, no. 4 (2019): 72-82, doi: 10.7546/nntdm.2019.25.4.72-82.

Cobb, Sarah C., Michelle L. Knox, Marcos Lopez, Terry McDonald and Patrick Mitchell. “Distribution of Constant Terms of Irreducible Polynomials in ℤ_{p}[*x*].” Notes on Number Theory and Discrete Mathematics 25.4 (2019): 72-82. Print, doi: 10.7546/nntdm.2019.25.4.72-82.