Visual Aspects of Gaussian Periods and Analogues

Abstract

Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties. In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations -- one relating to supercharacter theory, the other relating to class field theory -- along with discussions and visual examples of each. We end the paper by including some code for readers to generate images on their own.