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

We study the number N γ ( n , c , q ) of irreducible polynomials of degree n over F q 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 N γ ( n , c , q ) varying c and γ . We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any N γ ( n , c , q ) when n = a ( q − 1 ) for a nonzero constant term c and a nonzero trace γ . 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 .