Idele Class group

The idele class group of a number field \(K\) is defined as the following quotient. \[\begin{equation} \mathop{\mathrm{\mathfrak{C}}}(K) = \mathbb{A}_K^{\times} / K^{\times}. \end{equation}\]

Here \(\mathbb{A}_K^{\times}\) is the restricted product of \(\prod_{v \in M_K} K_v^{\times}\) where \(M_K\) is the set of all places \(v\) of \(K\) and \(K_v\) is the \(v\)-adic completion of \(K\) at \(v\).

The idele class group is not compact and is not a finite dimensional manifold. It is however a locally compact Abelian group.