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

We study the number Nγ(n, c, q) of irreducible polynomials of degree n over Fq where the trace γ and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of Nγ(n, c, q) varying c and γ. We show with concrete examples how our results improve previous known bounds. In addition, we improve upper and lower bounds of any Nγ(n, c, q) when n = a(q − 1) for nonzero constant term c and nonzero trace γ. As a byproduct, we give a simple and explicit formula for the number N(n, c, q) of irreducible polynomials over Fq of degree n = q − 1 with prescribed primitive constant term c.