Rings of algebraic integers have a number of well-known properties that make them particularly nice contexts in which to work. Among them is the fact that one can determine the elasticity of the ring simply by looking at the ideal class group. In this talk, we will explore how we might infer factorization properties of orders in algebraic number fields from specific relationships they might have with the corresponding number rings. We will also explore ways in which we might find orders which have (or fail to have) these relationships.

When it comes to mathematical structures such as groups or linear orders, we often consider them up to isomorphism. Determining if two structures are isomorphic usually relies on studying their structural properties, and so we can ask the following questions: is there a way to state these properties as formal sentences in some fixed language, and if we have models which satisfy a set of formal sentences, are those models necessarily isomorphic to each other? In this talk, we will briefly cover some basic ideas from model theory and the Scott Isomorphism theorem, which answers the second question above for countable structures.