On irreducible polynomials over finite fields
Number Theory
2012-10-16 v2 Combinatorics
Abstract
For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence N_{n+1}(q)^{1/(n+1)}/N_n(q)^{1/n} (n>=5.835*10^{14}) is strictly decreasing. We also prove that if q>8 then N_{n+1}(q)/N_n(q) (n=1,2,3,...) is strictly increasing.
Cite
@article{arxiv.1210.1562,
title = {On irreducible polynomials over finite fields},
author = {Zhi-Wei Sun},
journal= {arXiv preprint arXiv:1210.1562},
year = {2012}
}
Comments
7 pages. Add a result on N_{n+1}(q)/N_n(q) and its proof