For questions on groups and rings of adeles, self-dual topological rings built on an algebraic number field.

The ring of adeles is a self-dual topological ring built on the field of rational numbers (or, more generally, any algebraic number field). The ring of adeles allows one to elegantly describe the Artin reciprocity law, which is a vast generalization of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that $G$-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group $G$.

Further reference: Adele ring