A Generalization of Wantzel's Theorem, m-sectable angles, and the density of certain Chebyshev-polynomial images

TL;DR

This paper extends Wantzel's theorem to all Chebyshev polynomials, investigates the algebraic nature of m-sectable angles, and studies the density of their cosine values within algebraic number fields.

Contribution

It generalizes Wantzel's theorem to all Chebyshev polynomials and introduces a density measure for m-sectable angles in algebraic fields.

Findings

m-Sect contains only algebraic numbers when m is not a power of two

The density of m-Sect in algebraic number fields is zero for non-power-of-two m

Generalized Wantzel theorem applies to all Chebyshev polynomials

Abstract

The eponymous theorem of P.L. Wantzel presents a necessary and sufficient criterion for angle trisectability in terms of the third Chebyshev polynomial $T_3$, thus making it easy to prove that there exist non-trisectable angles. We generalize this theorem to the case of all Chebyshev polynomials $T_m$ . We also study the set \textbf{m-Sect} consisting of all cosines of $m$-sectable angles (see \S 1), showing that, when $m$ is not a power of two, \textbf{m-Sect} contains only algebraic numbers . We then introduce a notion of density based on the diophantine-geometric concept of height of an algebraic number and obtain a result on the density of certain polynomial images. Using this in conjunction with the Generalized Wantzel Theorem, we obtain our main result: for every real algebraic number field $K$, the set \textbf{m-Sect}\ $\cap\ K$ has density zero in $[-1,1]\ \cap\ K$ when $m$ is not a power of two. (To appear in the Journal of Pure and Applied Algebra.)