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.

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Keywords Finite fields, Irreducible polynomials, Trace and constant terms
Persistent URL dx.doi.org/10.1016/j.disc.2009.12.006
Journal Discrete Mathematics
Omidi Koma, B., Panario, D., & Wang, Q. (2010). The number of irreducible polynomials of degree n over Fq with given trace and constant terms. Discrete Mathematics, 310(8), 1282–1292. doi:10.1016/j.disc.2009.12.006