Arithmetic from an advanced perspective: an introduction to the Adeles

Abstract

Here we offer an introduction to the adele ring over the field of rational numbers Q and highlight some of its beautiful algebraic and topological structure. We then apply this rich structure to revisit some ancient results of number theory and place them within this modern context as well as make some new observations. We conclude by indicating how this theory enables us to extend the basic arithmetic of Q to the more subtle, complicated, and interesting setting of an arbitrary number field.