A generalization of the Hansen–Mullen conjecture on irreducible polynomials 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 which states that for n⩾3, there exist irreducible polynomials over Fq of degree n, with any one coefficient prescribed to any element of Fq (this being nonzero when the constant coefficient is being prescribed) and was proved by Wan. We introduce a variation of Wanʼs method to give restrictions subject to which this result can be extended to more than one prescribed coefficient; for example we show the asymptotical existence of irreducible polynomials with trace and any other one coefficient prescribed to any value. It also follows from our generalization the existence of irreducible polynomials with sequences of consecutive zero coefficients.