Distribution of constant terms of irreducible polynomials in ℤ_p[x]

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
Full paper (PDF, 206 Kb)

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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

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