Exponential sums are important tools in number theory for solving problems involving integers—and real numbers in general—that are often intractable by other means. Analogous sums can be considered in the framework of finite fields and turn out to be useful in various applications of finite fields.

A basic role in setting up exponential sums for finite fields is played by special group homomorphisms called characters. It is necessary to distinguish between two types of characters—namely, additive and multiplicative characters—depending on whether reference is made to the additive or the multiplicative group of the finite field. Exponential sums are formed by using the values of one or more characters and possibly combining them with weights or with other function values. If we only sum the values of a single character, we speak of a character sum.

In Section 1 we lay the foundation by first discussing characters of finite abelian groups and then specializing to finite fields. Explicit formulas for additive and multiplicative characters of finite fields can be given. Both types of characters satisfy important orthogonality relations.

Section 2 is devoted to Gaussian sums, which are arguably the most important types of exponential sums for finite fields as they govern the transition from the additive to the multiplicative structure and vice versa. They also appear in many other contexts in algebra and number theory. As an illustration of their usefulness in number theory, we present a proof of the law of quadratic reciprocity based on properties of Gaussian sums.