Cyclotomic polynomials at roots of unity

TL;DR

This paper develops explicit formulas for evaluating cyclotomic polynomials at roots of unity using finite Fourier analysis, providing new proofs, formulas, and computational methods for properties of these polynomials.

Contribution

It introduces new explicit evaluation formulas for cyclotomic polynomials at roots of unity and applies these to derive results on their coefficients and resultants.

Findings

Explicit formulas for $\

Evaluation of $\

Short computation method for the resultant of cyclotomic polynomials

Abstract

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $\Phi_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $\Phi_n(e^{2\pi i/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\Phi_n(x)$. We also obtain a formula for $\Phi_n'(e^{2\pi i/m}) / \Phi_n(e^{2\pi i/m})$ with $n \ne m$, which is effectively applied to $m \in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.