Roots of Chebyshev Polynomials: a purely algebraic approach

TL;DR

This paper provides an algebraic approach to understanding the roots of Chebyshev polynomials, proving their simplicity, location within (-1,1), and their irrationality through two methods.

Contribution

It introduces a purely algebraic method to analyze Chebyshev polynomial roots, offering new proofs of their properties.

Findings

Roots are simple and lie in (-1,1)

Most roots are irrational

Two algebraic proofs of root properties

Abstract

By using purely algebraic tools, we establish well-known properties of roots of Chebyshev polynomials. Especially, we show that these zeros are simple and lie in $(-1,1)$ and we prove in two ways that they are mostly irrational.