Irreducible factorization of translates of reversed Dickson polynomials over finite fields

TL;DR

This paper characterizes the irreducible factors of certain polynomials over finite fields, specifically those related to translates of reversed Dickson polynomials, providing conditions for their irreducibility.

Contribution

It offers a complete description of the irreducible factors of polynomials of the form g_s(y) over finite fields and establishes criteria for their irreducibility.

Findings

Explicit factorization over finite fields

Necessary and sufficient conditions for irreducibility

Connection to reversed Dickson polynomials

Abstract

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n = (q+1)/2$. For each $s \in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y): = y^n + (1-y)^n -s$, and we give a necessary and sufficient condition on $s$ for $g_s(y)$ to be irreducible.