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  "body": "\n## Summary\n\nThe adele ring $\\mathbb{A}_K$ of a number field $K$ is a restricted product of the completions $K_v$ of $K$ at all its [[places|BT5V3TSNY6]].\nThe restriction is taken with respect to the rings of integers of the non-archimedean completions.\nIt is a locally compact topological ring in which $K$ sits diagonally as a discrete subring with compact quotient.\nIt allows all completions of $K$ to be handled at once.\nIt is the setting of Tate's thesis, of the [[idele class group|9GHD9MXNYV]] form of [[class field theory|BEKCEYHCHX]] and of the theory of automorphic forms.\n\n## Explanations\n\n### L1 human\n\nA rational number can be seen inside the real numbers and inside each field of [[p-adic numbers|BT6T702C28]] $\\Q_p$.\nAn adele of $\\Q$ is a family with one real number and one $p$-adic number for each prime $p$.\nAll but finitely many of the $p$-adic entries must be $p$-adic integers.\nFor example, $1/6$ gives the adele whose entries are all $1/6$; it is allowed, since $1/6$ is a $p$-adic integer for every $p$ except $2$ and $3$.\nWorking with all these entries together turns many local statements into one global statement.\n",
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