# On the Middle Coefficient of a Cyclotomic Polynomial

## TL;DR

This paper presents an elementary proof that the middle coefficient of cyclotomic polynomials is either zero or an odd integer, depending on whether n is a power of two, clarifying a key property of these polynomials.

## Contribution

It offers a simple, elementary proof of a specific property of cyclotomic polynomial coefficients, improving understanding of their structure.

## Key findings

- Middle coefficient is zero if n is a power of 2.
- Middle coefficient is an odd integer for other n ≥ 3.
- Provides a clearer proof of a known property.

## Abstract

In this article, we provide a short and elementary proof of the following result: For $n \geq 3$ the middle coefficient of $\Phi_n(x)$ is either zero (when $n$ is a power of $2$) or an odd integer.