Course Objectives: This course introduces the subject of algebraic number theory from a modern (i.e., adelic) perspective. The goal is to present the material using tools which lend themselves to generalizations of the study of the ring of integers of a number field to the study of the arithmetic of orders in noncommutative (e.g., quaternion) algebras. Even for students who have seen an introduction to algebraic number theory from a classical (global) perspective, much of this material will either be new, or provide a new perspective on familiar material.
This course presumes a knowledge of Galois theory and metric topology (compactness, connectedness, etc). In addition, very basic properties of topological groups as well as Haar measure on locally compact abelian groups will be used, but if those notions are unfamiliar, they are easily picked up as we progress through the course.
The course will begin traditionally by defining the ring of integers of a number field and proving it is a Dedekind domain. Then there is a discussion of valuations, completions of a number field with respect to these valuations, and introduction of the adeles and ideles. Classical results of finiteness of the class number, Dirichlet's unit theorem, and product formula for valuations are all derived adelically. Then a detailed study of local fields is developed including the local-global correspondence and local ramification theory. Finally the arithmetic of global (e.g., number) fields is developed all through the lens of the local-global correspondence.