### 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.