Abstract. D. Wan very recently proved an asymptotic version of a conjecture of Hansen and Mullen concerning the distribution of irreducible polynomials over finite fields. In this note we prove that the conjecture is true in general by using machine calculation to verify the open cases remaining after Wan’s work. For a prime power q let Fq denote the finite field of order q. Hansen and Mullen in [4, p. 641] raise Conjecture B. Let a ∈ Fq and let n ≥ 2 be a positive integer. Fix an integer j with 0 ≤ j<n.Then there exists an irreducible polynomial f(x) =xn+ ∑n−1 k k=0 akx over Fq with aj = a except when (B1) q arbitrary and j = a =0; (B2) q =2m,n=2,j =1,and a =0
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