Distribution of irreducible polynomials of small degrees over finite fields

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=0 akxk over Fq with aj = a except when (B1) q arbitrary and j = a = 0; (B2) q = 2m , n = 2, j = 1, and a = 0. Clearly (B1) must be an exception, for otherwise f(x) is divisible by x. As for (B2), in characteristic two every element of Fq is a square, and so x2 +a0 = (x+b)2 is reducible. Using character sum estimates, in [6, Cor. 5.8] Wan provides an asymptotic version of Conjecture B by proving:

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: Kie H. Ham, Gary L. Mullen
• Keywords: Asymptotic Version, Conjecture, Irreducible Polynomial, MOC 1998