The theory of indices modulo n.

Abstract

This thesis is intended for an audience familiar with basic elementary number theory and acquainted with some abstract algebra: it discusses the theory of indices modulo n. Chapter 1 presents definitions and theorems from elementary number theory that form a background for the rest of this paper. Because primitive roots playa crucial role in the study of indices, we discuss primitive roots in Chapter 2. In that chapter, we establish which moduli possess primitive roots. The objective of Chapter 3 is the study of scalar indices and their properties. We find that in both theory and application, indices are analogous to logarithms. This analogy is emphasized and is used as a motivation to introduce certain results. Applications of scalar indices in solving various types of congruence equations is discussed and illustrated. Also included is the construction of a modular slide rule based on index theory. Chapter 4 extend the theory as developed in the previous chapters to make it applicable to arbitrary moduli, this by means of vector indices. Again, applications are given. Chapter 5 takes a more general approach to the topic and looks at the theory of indices from an algebraic point of view. This approach leads in a natural way to the generalization of indices to finite cyclic groups as well as to the direct product of two finite cyclic groups. We conclude by suggesting a direction for further study of this topic.