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  "body": "\n## Definition\n\nLet $K$ be a [[number field|MWBDFS2KJ5]].\nFor each [[place|BT5V3TSNY6]] $v$ of $K$, let $K_v$ be the completion of $K$ at $v$.\nFor each finite place $v$, let $\\mathcal{O}_v$ be the ring of integers of $K_v$ and $\\mathcal{O}_v^\\times$ its group of units.\nThe *idele group* $I_K$ is the set of families $x = (x_v)_v$ such that:\n\n1. $x_v \\in K_v^\\times$ for every place $v$;\n2. $x_v \\in \\mathcal{O}_v^\\times$ for all but finitely many finite places $v$.\n\nIt is a group under componentwise multiplication.\nIts topology has as a basis the sets $\\prod_{v \\in S} U_v \\times \\prod_{v \\notin S} \\mathcal{O}_v^\\times$.\nHere $S$ is a finite set of places containing all infinite places, and each $U_v$ is open in $K_v^\\times$.\nThe map $a \\mapsto (a, a, \\ldots)$ embeds $K^\\times$ in $I_K$ as a discrete subgroup.\nThe *idele class group* of $K$ is the [[quotient group|0KYX2N69J0]]\n$$C_K = I_K / K^\\times,$$\nwith the quotient topology.\n\n## Remarks\n\nExample: for $K = \\Q$, the family with $x_\\infty = 2$, $x_3 = 1/3$ and $x_p = 1$ for all other $p$ is an idele, because only finitely many entries are not units.\nNon-example: the family with $x_p = p$ for every prime $p$ and $x_\\infty = 1$ is not an idele, because $x_p \\notin \\Z_p^\\times$ for infinitely many $p$.\nThe ideles are the units of the [[adele ring|SEH68BA1QQ]] $\\mathbb{A}_K$, but the topology above is finer than the subspace topology from $\\mathbb{A}_K$.\nThe idele norm $\\abs{x} = \\prod_v \\abs{x_v}_v$ equals $1$ on $K^\\times$ by the product formula, so it is defined on $C_K$.\nIts kernel $C_K^1$ is compact.\nThe [[ideal class group|WQ46BD3ARY]] of $K$ is a quotient of $C_K$.\nThe same construction works for every [[global field|634XKY8RZ0]].\n",
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    "name": "Idele class group",
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