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  "body": "\n## Summary\n\nA place of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a class of equivalent nontrivial absolute values on $K$.\nThe finite places correspond to the nonzero prime ideals of the ring of integers.\nThe infinite places correspond to the real embeddings and the pairs of complex conjugate embeddings of $K$.\nEach place gives a completion of $K$, which is a [[local field|J85FZEQG38]] or $\\R$ or $\\C$.\nThe places together are the index set of the [[adele ring|SEH68BA1QQ]].\nFor $K = \\Q$, [[Ostrowski's theorem|GCQ23XT6VD]] says that the places are the primes $p$ and one infinite place.\n\n## Explanations\n\n### L1 human\n\nA place is a way of measuring the size of the numbers of a field.\nOn $\\Q$ there is the usual size, and for each prime $p$ there is the [[p-adic size|WGN82TCEF2]].\nFor the $p$-adic size, a number is small when it is divisible by a high power of $p$.\nFor example, $\\abs{75}_5 = 1/25$, because $75 = 3 \\cdot 5^2$.\nNumber theorists treat all these ways of measuring on an equal footing, and the places are their list.\n",
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