A new identity of Dickson polynomials
Antonia Wilson Bluher · Finite Fields and Their Applications · 2022
A new polynomial identity is found for Dickson polynomials in characteristic 2. The identity is used to prove that the two polynomials x q + 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 = 2 n > 2 , and C ( x ) is a Müller–Cohen–Matthews polynomial of degree ( q 2 − q ) / 2 . In addition, a new proof is obtained of the known result that C ( x ) induces a permutation on F 2 m if and only if 2 m and n are relatively prime.