Polynomial sequences related to Chebyshev polynomials and the minimal polynomial of $2\cos (2\pi /n)$

TL;DR

This paper introduces polynomial sequences related to Chebyshev polynomials to efficiently compute the minimal polynomial of 2cos(2π/n), improving previous methods and providing explicit tables for n ≤ 120.

Contribution

It presents a new approach to calculate minimal polynomials of 2cos(2π/n) using polynomial sequences related to Chebyshev polynomials, avoiding recursion.

Findings

Derived polynomial sequences with the same recurrence as Chebyshev polynomials.

Established divisibility properties of these polynomials related to minimal polynomials.

Provided explicit tables of minimal polynomials for n ≤ 120.

Abstract

In this paper we consider the minimal polynomial $\psi_n(x)$ of $2\cos (2\pi /n)$. We introduce some polynomial sequences with the same recurrence relation as the rescaled Chebyshev polynomials $t_n(x)=2\, T_n(x/2)$ of the first kind, which turn out to be related to those of various kinds, all coming from those of the second kind. We see that $t_n(x)\pm 2=2(T_n(x/2)\pm 1)$ are divisible by the square of either of these polynomials. Then by appropriately removing unnecessary factors from these polynomials, we can easily calculate $\psi_n(x)$ without recursion, which improves Barnes' result in 1977. As an appendix, we give a compact table of the minimal polynomials $\psi_n(x)$ of $2\cos (2\pi /n)$ for $n\leqslant 120$.