idèle

Let $K$ be a number field. For each finite prime $v$ of $K$, let ${𝔬}_v$ be the valuation ring of the completion $K_v$ of $K$ at $v$, and let $U_v$ be the group of units in ${𝔬}_v$. Then each group $U_v$ is a compact open subgroup of the group of units $K_v^{*}$ of $K_v$. The idèle group ${𝕀}_K$ of $K$ is defined to be the restricted direct product of the multiplicative groups $\{K_v^{*}\}$ with respect to the compact open subgroups $\{U_v\}$, taken over all finite primes and infinite primes $v$ of $K$.

The units $K^{*}$ in $K$ embed into ${𝕀}_K$ via the diagonal embedding

$$x\mapsto \prod _v{x}_v,$$

where $x_v$ is the image of $x$ under the embedding $K\hookrightarrow K_v$ of $K$ into its completion $K_v$. As in the case of adèles, the group $K^{*}$ is a discrete subgroup of the group of idèles ${𝕀}_K$, but unlike the case of adèles, the quotient group ${𝕀}_K/K^{*}$ is not a compact group. It is, however, possible to define a certain subgroup of the idèles (the subgroup of norm 1 elements) which does have compact quotient under $K^{*}$.

Warning: The group ${𝕀}_K$ is a multiplicative subgroup of the ring of adèles ${𝔸}_K$, but the topology on ${𝕀}_K$ is different from the subspace topology that ${𝕀}_K$ would have as a subset of ${𝔸}_K$.