Date of Award

6-2011

Document Type

Open Access

Degree Name

Bachelor of Science

Department

Mathematics

First Advisor

Karl Zimmermann

Language

English

Keywords

ideals, prime, extension, integers, Galois theory, algebra

Abstract

It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that prime ideals of OK need not remain prime when they are extended into OL; instead, they can split into a product of prime ideals of OL in a very structured way. If L is a normal extension of K, we can use Galois theory to further study this splitting by considering the intermediate fields of K and L, as well as quotient rings of the associated rings of integers. In this paper, we will introduce these topics of algebraic number theory, prove that unique factorization of ideals holds using two different methods, and observe the patterns that arise in the splitting of prime ideals.

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