The last chapter of the Disquisitiones of Gauss

TL;DR

This paper reviews Gauss's assertions and proofs regarding the construction of regular polygons with a given number of sides, focusing on roots of unity and historical context in the last chapter of Disquisitiones Arithmeticae.

Contribution

It clarifies Gauss's claims and proofs about constructible polygons, providing historical insights and classical references that are often overlooked.

Findings

Gauss proved the constructibility of certain regular polygons.

Historical analysis of Gauss's original assertions and proofs.

Compilation of classical references related to Gauss's work.

Abstract

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.