Wantzel定理的推广、m-可截角和某些切比雪夫多项式象的密度
A Generalization of Wantzel's Theorem, m-sectable angles, and the density of certain Chebyshev-polynomial images
摘要：
P.L. Wantzel利用第三类Chebyshev多项式$T_3$给出了角可三等分的一个充要条件，从而使证明存在不可三等分角变得容易.我们将这个定理推广到所有切比雪夫多项式$T_m$的情形。我们还研究了由$m$可截角的所有余弦组成的集合\textbf{m-Sect}（见§1），表明当$m$不是2的幂时，\textbf{m-Sect}只包含代数数。然后，我们介绍了密度的概念的基础上的丢番图几何概念的高度的代数数，并得到一个结果的密度的某些多项式图像。结合广义Wantzel定理，我们得到了我们的主要结果：对于每个实代数数域$K$，当$m$不是2的幂时，集合\textbf{m-Sect} \ $\cap\ K$在$[-1,1]\ \cap\ K$中密度为零。(To发表在Journal of Pure and Applied Algebra上。
英文摘要：
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 §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.)
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