Andrea Bodkin

Document Type

Honors Project

First Advisor

Dr. Melissa Hoover

Degree Award Date

Spring 2006

Keywords

Abstract algebra, group theory, rings, fields

Disciplines

Algebra | Mathematics

Abstract

Abstract algebra is the branch of mathematics concerned with the study of algebraic structures, the most important of which are groups, rings, and fields, where rings are an extension of groups and fields an extension of rings. These three topics provide the theoretical foundation necessary for the study of algebra, which includes everything from solving equations for x and factoring polynomials in high school math to applications involving check digit schemes and cryptography. This project is an independent study of rings and fields. I chose to do this project because, while groups are studied in MATH 400, Modem Algebra, there is not enough time in one semester to cover rings and fields also. Knowledge of rings and fields, however, is important for anyone entering a graduate program in mathematics.

This paper is intended to be accessible for anyone with some knowledge of group theory. It begins with a review to remind the reader of the concepts and definitions of groups most necessary for the study of rings and fields, and then moves into these two topics, showing their relationship to groups and to one another, along with many examples of how and for what they can be used.

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