Dickson Polynomials and Irreducible Polynomials Over Finite Fields

Abstract

In this paper we establish necessary and sufficient conditions for D n ( x, a) + b to be irreducible over F q , where a, b ∈ F q , the finite field F q of order q, and D n ( x, a) is the Dickson polynomial of degree n with parameter a ∈ F q . As a consequence we construct several families of irreducible polynomials of arbitrarily high degrees. In addition we show that the following conjecture of Chowla and Zassenhaus from 1968 is false for Dickson polynomials: if ƒ( x) has integral coefficients and has degree at least two, and p is a sufficiently large prime for which ƒ( x) does not permute F p , then there is an element c ∈ F p so that ƒ( x) + c is irreducible over F p . We also show that the minimal polynomials of elements which generate one type of optimal normal bases can be derived from Dickson polynomials.