A note on the Brawley-Carlitz theorem on irreducibility of composed products of polynomials over finite fields

TL;DR

This paper presents a new proof of the Brawley-Carlitz theorem on polynomial irreducibility over finite fields, removing the need for associativity and exploring conditions based on polynomial degrees.

Contribution

It offers a novel proof that does not rely on associativity and provides degree-based criteria for irreducibility of composed polynomial products.

Findings

New proof of the Brawley-Carlitz theorem

Associativity is not necessary for the composed product operation

Degree conditions influence polynomial irreducibility

Abstract

We give a new proof of the Brawley-Carlitz theorem on irreducibility of the composed products of irreducible polynomials. Our proof shows that associativity of the binary operation for the composed product is not necessary. We then investigate binary operations defined by polynomial functions, and give a sufficient condition in terms of degrees for the requirement in the Brawley-Carlitz theorem.