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 $\A_K$