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

TL;DR

The paper develops a nonrecursive framework for the minimal polynomials $ψ_n(x)$ of $2\cos(2\pi/n)$ by linking them to rescaled Chebyshev polynomials and new polynomial families $c_n(x)$, $p^ abla_n(x)$, and $q^ abla_n(x)$. It provides explicit, parity-dependent formulas for $ψ_{m/n}(x)$ in terms of divisor-structured products and demonstrates divisibility properties that enable direct computation, improving Barnes's 1977 results without using cyclotomic polynomials. The approach unifies Chebyshev-related polynomials of several kinds and yields practical expressions for $ψ_n(x)$, along with an appendix tabulating cases up to $n\le120$. This work enhances both theoretical understanding and computational accessibility of minimal polynomials associated with $2\cos(2\pi/n)$ and related angles.

Abstract

In this paper we consider the minimal polynomial $ψ_n(x)$ of $2\cos (2π/n)$. 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 $ψ_n(x)$ without recursion, which improves Barnes' result in 1977. As an appendix, we give a compact table of the minimal polynomials $ψ_n(x)$ of $2\cos (2π/n)$ for $n\leqslant 120$.

Theorems & Definitions (25)

• Theorem 1.1: WZ93, p. $473$, Lemma
• Theorem 1.2: WZ93, p. $471$, Theorem
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6: Bar77, Theorem $11$
• Corollary 1.7
• Example 1.8
• Proposition 2.1
• proof