Yet another proof from the Book: the Gauss theorem on regular polygons

TL;DR

This paper provides a concise, elementary proof of Gauss's theorem on the constructibility of regular polygons, making the classical result more accessible and less reliant on complex background material.

Contribution

It offers a simplified, well-motivated proof of Gauss's theorem, enhancing understanding for students and professionals alike.

Findings

Elementary proof of Gauss's theorem presented

Proof is accessible for students with basic polynomial and complex number knowledge

Enhances clarity and motivation of the classical result

Abstract

This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated material and are far from being economic. In this note a short elementary proof of the Gauss theorem is presented. The note is accessible for students familiar with polynomials and complex numbers, and could be an interesting easy reading for professional mathematicians.