On diamond products ensuring irreducibility of the associated composed product

Abstract

Let f and g be two irreducible polynomials of coprime degrees m and n whose zeroes lie in a set $G\subseteq {\overline{\mathbf{F}}}_q$. Let $\diamond$ be a diamond product on G. We define the weaker cancelation property of $\diamond$ and show that it is sufficient to conclude that the composed product of f and g derived from $\diamond$ is an irreducible polynomial of degree mn. We also prove that a wide class of diamond products on finite fields satisfy the weaker cancelation property. These results extend the corresponding results of Brawley and Carlitz (1987), and Munemasa and Nakamura (2016).

Keywords:

• Composed product
• diamond product
• irreducible polynomial
• permutation polynomial

2020 Mathematics Subject Classification:

• 11T06

Acknowledgments

We would like to thank the anonymous referee for the valuable suggestions and comments, which improved the paper.