On the Existence of Irreducible Polynomials with Prescribed Coefficients over Finite Fields

Let q be a prime power and Fq the finite field with q elements. We examine the existence of irreducible polynomials with prescribed coefficients over Fq . We focus on a conjecture by Hansen and Mullen [25, Conjecture A] which states that for n ≥ 3, there exist irreducible polynomials over Fq with any one coefficient prescribed to any element of Fq (this being nonzero when the constant coefficient is being prescribed) and was proven by Wan [42]. We introduce a variation of Wan's method to give restrictions subject to which this result can be extended to more than one prescribed coefficient. It also follows from our generalization the existence of irreducible polynomials with sequences of consecutive zero coefficients.