Trigonometric Sums and Characters

Abstract

Let m be a positive integer. We have seen that the set of integers can be partitioned into residue classes

$$A_0 ,A_1 , \ldots ,A_{m - 1}$$

where A _s is the set of integers congruent to s mod m. We can define the operation of addition on these residue classes by

$${A_s} + {A_t} = {A_u},{\text{ }}u = \left\{ {_{s + t - m{\text{ }}}^{s + t}} \right._{{\text{if }}s{\text{ + t}} \geqslant {\text{m}}{\text{.}}}^{{\text{if }}s + t < m,}$$

This definition satisfies properties associated with groups. Within the theory of groups there is a representation theory whereby more abstract objects are given concrete representations, and this theory has very useful applications (for example, in electronics). In this section we discuss the method of representing residue classes which form an additive group.