书中的更多证明：根式中方程的可解性与不可解性
Some more proofs from the Book: solvability and insolvability of equations in radicals
AI总结：
本文通过初等方法证明了正则多边形可构造性、三次方程不可用实根式解以及五次方程不可用复根式解等经典结果。
摘要：
本文纯属阐述性。我们给出了以下三个著名结果的简短初等证明：* 正则多边形可构造性的高斯定理；* 无法用实根式解的三次方程的存在性；* 无法用复根式解的五次方程的存在性（伽罗瓦定理）。这些结果的陈述简单而广为人知。然而，大多数教科书中的证明依赖于大量未经动机的材料，远非经济高效。我们不使用术语‘伽罗瓦群’或甚至‘群’。本文适合熟悉多项式和复数的学生，也可能是专业数学家的有趣简读材料。简短的英文版本后附有扩展的俄文版本，其中在呈现证明之前，通过一系列带有提示或解答的问题来阐述主要思想。
英文摘要：
This paper is purely expository. We present short elementary proofs of * the Gauss Theorem on constructibility of regular polygons; * the existence of a cubic equation unsolvable in real radicals; * the existence of a quintic equation unsolvable in complex radicals (Galois Theorem). The statements of these celebrated results are simple and well-known. However, their proofs given in most textbooks rely upon much unmotivated material and are far from being economic. We do not use the terms `Galois group' or even `group'. The paper is accessible for students familiar with polynomials and complex numbers, and could be an interesting easy reading for professional mathematicians. Short English version is followed by an extended Russian version where before presenting the proofs we illustrate the main ideas by sequences of problems with hints or solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
