Divisibility criteria and coefficient formulas for cyclotomic polynomials

TL;DR

This paper develops divisibility criteria for cyclotomic polynomials, introduces new coefficient formulas involving Ramanujan sums, and explores recursive relations between coefficients of related cyclotomic polynomials.

Contribution

It provides necessary and sufficient divisibility conditions, new coefficient formulas, and recursive relations, advancing understanding of cyclotomic polynomial properties.

Findings

Established divisibility criteria for cyclotomic polynomials

Derived new coefficient formulas involving Ramanujan sums

Proposed recursive relations between related cyclotomic polynomial coefficients

Abstract

We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the coefficients of cyclotomic polynomials and we propose a recursive relation between the coefficients of two cyclotomic polynomials whose indexes differ by a prime factor.