On irreducible polynomials over finite fields

TL;DR

This paper investigates the asymptotic behavior of the number of monic irreducible polynomials over finite fields, establishing monotonicity properties of related sequences for large degrees and field sizes.

Contribution

It proves new monotonicity results for sequences derived from counts of irreducible polynomials over finite fields, extending understanding of their growth patterns.

Findings

N_n(q)^{1/n} is strictly increasing for large n

The ratio N_{n+1}(q)^{1/(n+1)}/N_n(q)^{1/n} is decreasing for very large n

If q>8, then N_{n+1}(q)/N_n(q) is strictly increasing

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.