A new identity of Dickson polynomials

Abstract

A new polynomial identity is found for Dickson polynomials in characteristic 2. The identity is used to prove that the two polynomials xq+1+x+1/a and C(x)+a have the same splitting fields over F, where F is a field of characteristic 2, 0≠a∈F, q=2n>2, and C(x) is a Müller–Cohen–Matthews polynomial of degree (q2−q)/2. In addition, a new proof is obtained of the known result that C(x) induces a permutation on F2m if and only if 2m and n are relatively prime.