Date of Award

6-2013

Document Type

Open Access

Degree Name

Bachelor of Science

Department

Mathematics

First Advisor

Karl Zimmerman

Language

English

Keywords

abstract algebra, number theory, arithmetic

Abstract

Though it may seem non-intuitive, abstract algebra is often useful in the study of number theory. In this thesis, we explore some uses of abstract algebra to prove number theoretic statements. We begin by examining the structure of unique factorization domains in general. Then we introduce number fields and their rings of algebraic integers, whose structures have characteristics that are analogous to some of those of the rational numbers and the rational integers. Next we discuss quadratic fields, a special case of number fields that have important applications to number theoretic problems. We will use the structures that we introduce throughout the thesis to prove several number theoretic statements, including the Fundamental Theorem of Arithmetic, Fermat’s Theorem on Sums of Squares, and the Ramanujan-Nagell Theorem, as well as to generate a myriad of other interesting tangentially related results.

Download

Included in

Algebra Commons

Share

COinS