Dickson Polynomials and Irreducible Polynomials Over Finite Fields Original Research Article

In this paper we establish necessary and sufficient conditions for Dn(x, a) + b to be irreducible over Fq, where a, b set membership, variant Fq, the finite field Fq of order q, and Dn(x, a) is the Dickson polynomial of degree n with parameter a set membership, variant Fq. 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 Fp, then there is an element c set membership, variant Fp so that ƒ(x) + c is irreducible over Fp. We also show that the minimal polynomials of elements which generate one type of optimal normal bases can be derived from Dickson polynomials.