Definition 2.2.1. The ring of adeles of \(\mathbb{Q}\) is the subring \[\mathbb{A}= \Big\{ (x_\infty, x_2, x_3, x_5, \ldots) \in \mathbb{R}\times \prod\nolimits_p \mathbb{Q}_p \ :\ x_p \in \mathbb{Z}_p \text{ for all but finitely many } p \Big\}\] of the product ring. It is given the unique topology of a topological ring for which \(\mathbb{R}\times \prod_p \mathbb{Z}_p\), with its product topology, is an open subring. The field \(\mathbb{Q}\) is embedded in \(\mathbb{A}\) diagonally.