A Generalization of Wantzel’s Theorem, M-sectable Angles, and the Density of Certain Chebyshev-polynomial Images
نویسندگان
چکیده
The eponymous theorem of P.L. Wantzel [Wan] presents a necessary and sufficient criterion for angle trisectability in terms of the third Chebyshev polynomial T3, thus making it easy to prove that there exist non-trisectable angles. We generalize this theorem to the case of all Chebyshev polynomials Tm (Corollary 1.4.1). We also study the set m-Sect consisting of all cosines of m-sectable angles (see §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 is has density zero in [−1, 1] ∩ K when m is not a power of two (Corollary 1.5.1).
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