Variations on a question concerning the degrees of divisors of x^n-1

TL;DR

This paper investigates the frequency of divisors of x^n-1 across all degrees in polynomial rings over finite fields, extending previous work from integers to finite fields and exploring uniformity across primes.

Contribution

It introduces the study of divisors of x^n-1 in F_p[x] and examines conditions under which the divisor degree property holds for all primes p.

Findings

Analyzed divisibility properties of x^n-1 in finite fields.

Extended previous integer-based results to polynomial rings over finite fields.

Identified conditions for the property to hold uniformly across all primes.

Abstract

In this paper, we examine a natural question concerning the divisors of the polynomial x^n-1: "How often does x^n-1 have a divisor of every degree between 1 and n?" In a previous paper, we considered the situation when x^n-1 is factored in Z[x]. In this paper, we replace Z[x] with F_p[x], where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p.