Abstract: Unique factorization into prime numbers is a familiar and important property of integers. In this talk, we will examine rings of algebraic integers (number rings), some of which possess this property and some of which do not. Sorting out which ones have unique factorization, and among those that do not, how factorization does work, was the historical context for the development of the notion of ideals. We'll see how this works, and how ideals "save" unique factorization in number rings. Finally, we'll see how the amount of "saving" that has to be done is measured, in a sense, by the class number of an algebraic number field, we'll talk about how to calculate class numbers, and we'll prove that every number field has a finite class number.

No knowledge of number theory will be assumed, nor any algebra beyond basic familiarity with rings, fields, and ideals.