On the divisors of x^n-1 in F_p[x]

TL;DR

This paper investigates integers n for which the polynomial x^n - 1 over finite fields F_p[x] has divisors of all degrees up to n, showing under GRH that such n are rare with asymptotic density zero.

Contribution

It extends previous work from integers to finite fields and establishes that the set of such n has asymptotic density zero assuming GRH.

Findings

Under GRH, the set of integers n with divisors of all degrees up to n in F_p[x] has density zero.

Provides bounds and asymptotic analysis for these integers.

Connects divisor properties in polynomial rings over finite fields to number theoretic conjectures.

Abstract

In a recent paper, we considered integers n for which the polynomial x^n - 1 has a divisor in Z[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial x^n-1 has a divisor in F_p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.