Variations on a question concerning the degrees of divisors of xⁿ-1

2012/06/14 by Thompson, Lola
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1206.4355

Abstract

In this paper, we examine a natural question concerning the divisors of the polynomial xⁿ-1: "How often does xⁿ-1 have a divisor of every degree between 1 and n?" In a previous paper, we considered the situation when xⁿ-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.

Related

• On the divisors of xⁿ-1 in F_p[x]
• On integers n for which Xⁿ-1 has a divisor of every degree
• Polynomials with divisors of every degree
• On the degrees of divisors of Tⁿ-1
• On the coefficients of divisors of xⁿ-1