Gauss and Regular Polygons: Gaussian Periods

Introduction

In order to solve the equation $ z^{n} - 1 = 0$ Gauss introduced some sums of the $ n^{th}$ roots of unity which he called periods, and using these periods he was able to reduce the solution of $ z^{n} - 1 = 0$ to a sequence of solutions of equations of lower degrees. The technique offered by Gauss is extremely beautiful and completely novel and it uses the symmetry between the various $ n^{th}$ roots of unity to achieve the final solution.

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Gauss and Regular Polygons: Cyclotomic Polynomials

Introduction

The word "Cyclotomy" literally means "cutting a circle". So the subtitle of the post suggests that the post is going to be about some polynomials which are related to cutting a circle. Cutting a circle actually refers to dividing a given circle into a number of arcs of same length. Supposing that we are able to divide a given circle into, say $ n$, arcs of equal length by means of points $ P_{0}, P_{1}, \ldots, P_{n - 1}$ then joining the adjacent points we obtain a regular polygon $ P_{0}P_{1}\ldots P_{n - 1}$ of $ n$ sides. Therefore cyclotomic polynomials are somehow related to the construction of regular polygons.

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Gauss and Regular Polygons: Complex Numbers

Introduction to Complex Numbers

Complex numbers are not really complex! In fact they are reasonably simple to understand and operate upon. The concept is definitely strange on a first look, but is damn powerful and has diverse ramifications in various branches of mathematics. Now, to illustrate the point that these numbers are really simple, we are gonna define them in terms of quantities already known.

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