This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open problems are presented along the way, with particular emphasis on Gauss periods, Kloosterman sums and one variable exponential sums. One main tool is the applications of various p-adic methods. For this reason, the author has also included a brief exposition of certain p-adic estimates of exponential sums. The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University. Notes were taken by Shaoshi Chen and Ruichen Xu.

中文翻译：

---

有限域上的指数和

这是一篇关于有限域上指数和的代数方面的说明性论文。这是一个新的方向。一路上介绍了各种示例、结果和未解决的问题，特别强调高斯周期、Kloosterman 和和一个变量指数和。一个主要工具是各种 p-adic 方法的应用。为此，作者还简要介绍了指数和的某些p- adic 估计。该材料基于厦门大学2020年在线数论暑期学校的讲座。陈少士和徐瑞臣做笔记。