The number of irreducible polynomials of degree n over Fq with given trace and constant terms

We study the number N-gamma (n, c, q) of irreducible polynomials of degree n over F-q where the trace gamma and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of N-gamma (n, c, q) varying c and gamma. We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any N-gamma(n, c, q) when n = a(q - 1) for a nonzero constant term c and a nonzero trace gamma. As a byproduct, we give a simple and explicit formula for the number N(n, c. q) of irreducible polynomials over F-q of degree n = q-1 with a prescribed primitive constant term c. (C) 2009 Elsevier B.V. All rights reserved.