adèle

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 therestricted direct product of the collection of locally compactadditive groups $\{K_v\}$ over all primes $v$ of $K$ (both finiteprimes and infinite primes), with respect to the collection of compactopen 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. Theoriginal 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 underthe 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 thequotient group ${𝔸}_K/K$ is a compact space in the quotient topology.