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.
2.2 The adeles of the rational numbers
Throughout this section \(p\) denotes a prime number, \(\mathbb{Q}_p\) the field of \(p\)-adic numbers and \(\mathbb{Z}_p\) its ring of integers.
Lemma 2.2.2. Put \(D = [0, 1) \times \prod_p \mathbb{Z}_p \subset \mathbb{A}\). Every \(x \in \mathbb{A}\) can be written in a unique way as \begin{equation} \tag{2.2.2.1} x = \gamma + d, \qquad \gamma \in \mathbb{Q},\ d \in D. \end{equation}
Proof. See the reference given for Theorem 2.2.3.□
Theorem 2.2.3 (Discreteness and cocompactness). The subgroup \(\mathbb{Q}\subset \mathbb{A}\) is discrete, and the quotient \(\mathbb{A}/ \mathbb{Q}\) is compact.
Proof. By (2.2.2.1) the open neighbourhood \((-1, 1) \times \prod_p \mathbb{Z}_p\) of \(0\) meets \(\mathbb{Q}\) only in \(0\), and \(\mathbb{A}/ \mathbb{Q}\) is the image of the compact set \([0, 1] \times \prod_p \mathbb{Z}_p\).□
Remark 2.2.4. Definition 2.2.1 and Theorem 2.2.3 have analogues for every global field; see [Weil-AAG].