#### 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.

#### Recommended Citation

Bonventre, Peter J., "Factorization of Primes Primes Primes: Elements Ideals and in Extensions" (2011). *Honors Theses*. 944.

https://digitalworks.union.edu/theses/944