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

The eponymous theorem of P.L. Wantzel [5] presents a necessary and sufficient criterion for angle trisectability in terms of the third Chebyshev polynomial T3T3, thus making it easy to prove that there exist non-trisectable angles. We generalize this theorem to the case of all Chebyshev polynomials TmTm (Corollary 1.4.1). We also study the set m-Sect consisting of all cosines of m-sectable angles (see Section 1), showing that, when m is not a power of two, m-Sect contains only algebraic numbers (Theorem 1.1). 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 m-Sect ∩K has density zero in [−1,1]∩K when m is not a power of two (Corollary 1.5.1).