Irreducible factorization of translates of reversed Dickson polynomials over finite fields

Let F-q be a field of q elements, where q is a power of an odd prime. Fix n = (q + 1)/2. For each s is an element of F-q, we describe all the irreducible factors over F q 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. (C) 2018 Elsevier Inc. All rights reserved.