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

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