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

AbstractWe 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 the previously known bounds. In addition, we improve upper and lower bounds of any Nγ(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 Fq of degree n=q−1 with a prescribed primitive constant term c

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 the previously known bounds. In addition, we improve upper and lower bounds of any Nγ (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 Fq of degree n = q - 1 with a prescribed primitive constant term c