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

作者：Kahn Peter J^*

来源：Journal of Pure and Applied Algebra, 2015, 219(9): 3931-3945.

DOI：10.1016/j.jpaa.2014.12.029

The eponymous theorem of P.L. Wantzel [5] 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, (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 boolean AND K has density zero in [-1, 1] boolean AND K when m is not a power of two (Corollary 1.5.1).

• 出版日期2015-9