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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic 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 $\\mathcal{O}_v = \\set{x \\in K_v : \\abs{x}_v \\le 1}$ be its valuation ring.\nThe *adele ring* of $K$ is\n$$\\mathbb{A}_K = \\Big\\{ (x_v)_v \\in \\prod_v K_v : x_v \\in \\mathcal{O}_v \\text{ for all but finitely many finite } v \\Big\\},$$\nwith componentwise addition and multiplication, so it is a [[ring|0B1261B5ED]].\nIts topology is the restricted product topology.\nA basis of open sets is formed by the sets $\\prod_{v \\in S} U_v \\times \\prod_{v \\notin S} \\mathcal{O}_v$.\nHere $S$ is a finite set of places containing all infinite places, and each $U_v \\subseteq K_v$ is open.\n\n## Remarks\n\nWith this topology, $\\mathbb{A}_K$ is a [[locally compact|0FE2SB8WVJ]] topological ring.\nThe topology is finer than the one induced by the [[product topology|0GBG8B49AA]] on $\\prod_v K_v$.\nThe diagonal map $K \\to \\mathbb{A}_K$, $x \\mapsto (x)_v$, is well defined because an element of $K$ lies in $\\mathcal{O}_v$ for all but finitely many $v$; its image is discrete and $\\mathbb{A}_K / K$ is compact.\n\nExample: for $K = \\Q$, the places are $\\infty$ and the primes $p$, with $\\Q_\\infty = \\R$, $\\Q_p$ the [[p-adic numbers|BT6T702C28]] and $\\mathcal{O}_p = \\Z_p$ the [[p-adic integers|V6TJAFZ4MD]].\nThe family $x = (x_v)$ with $x_\\infty = \\pi$, $x_2 = 1/2$ and $x_p = 1$ for $p \\ge 3$ is an adele, since only one $p$-adic entry lies outside $\\Z_p$.\n\nNon-example: the family with $x_\\infty = 0$ and $x_p = 1/p$ for every prime $p$ lies in $\\prod_v \\Q_v$ but not in $\\mathbb{A}_{\\Q}$, since $1/p \\notin \\Z_p$ for infinitely many $p$.\n",
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  "msc": [
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  "name": "Adele ring",
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