Let $K$ be a number field. For each finite prime $v$ of $K$ , let $\o_v$ denote the valuation ring of the completion $K_v$ of $K$ at $v$ . The adèle group $\A_K$ of $K$ is defined to be the restricted direct product of the collection of locally compact additive groups $\{K_v\}$ over all primes $v$ of $K$ (both finite primes and infinite primes), with respect to the collection of compact open subgroups $\{\o_v\}$ defined for all finite primes $v$ .

The set $\A_K$ inherits addition and multiplication operations (defined pointwise) which make it into a topological ring. The original field $K$ embeds as a ring into $\A_K$ via the map $$ x \mapsto \prod_v x_v. $$ defined for $x \in K$ , where $x_v$ denotes the image of $x$ in $K_v$ under the embedding $K \hookrightarrow K_v$ . Note that $x_v \in \o_v$ for all but finitely many $v$ , so that the element $x$ is sent under the above definition into the restricted direct product as claimed.

It turns out that the image of $K$ in $\A_K$ is a discrete set and the quotient group $\A_K/K$ is a compact space in the quotient topology.