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  "body": "\n## Statement\n\nLet $\\abs{\\cdot}_*$ be a nontrivial absolute value on the field $\\Q$ of rational numbers.\nThen $\\abs{\\cdot}_*$ is equivalent either to the usual [[absolute value|0FFXKTKKQC]] $\\abs{\\cdot}_\\infty$ or to the [[p-adic absolute value|WGN82TCEF2]] $\\abs{\\cdot}_p$ for exactly one [[prime number|01826KMG2H]] $p$.\nHere two absolute values are equivalent when one is a positive real power of the other.\n\n## Remarks\n\nProof idea: suppose first that $\\abs{n}_* \u003e 1$ for some positive integer $n$.\nWriting integers in base $b$ and letting the number of digits grow, one shows $\\abs{n}_* = n^{\\alpha}$ for all positive integers $n$, with a fixed $0 \u003c \\alpha \\le 1$; so $\\abs{\\cdot}_* = \\abs{\\cdot}_\\infty^{\\alpha}$.\nOtherwise $\\abs{n}_* \\le 1$ for all integers $n$, and the integers with $\\abs{n}_* \u003c 1$ form a nonzero prime ideal $p\\Z$.\nThen $\\abs{x}_* = \\abs{p}_*^{v_p(x)}$, where $v_p$ is the [[p-adic valuation|03HDF50VN4]], and this is a power of $\\abs{x}_p$.\n\nThe absolute values $\\abs{\\cdot}_\\infty$ and $\\abs{\\cdot}_p$ for different $p$ are pairwise inequivalent: for instance $\\abs{p}_p \u003c 1$ while $\\abs{p}_q = 1$ for a prime $q \\ne p$ and $\\abs{p}_\\infty \u003e 1$.\nThe hypothesis of nontriviality is needed: the trivial absolute value, equal to $1$ on all $x \\ne 0$, is equivalent to none of these.\n\n## Explanations\n\n### L1 human\n\nThere are only a few reasonable ways to measure the size of a rational number.\nOne is the usual size; the others are the $p$-adic sizes, one for each prime $p$, which measure divisibility by $p$.\nUp to rescaling, there is nothing else.\n\n### L2 human\n\nThe theorem explains why number theory works with the real numbers and the [[p-adic numbers|BT6T702C28]] side by side: these are the completions of $\\Q$ for all its [[places|BT5V3TSNY6]].\nIt leads to the product formula $\\abs{x}_\\infty \\prod_p \\abs{x}_p = 1$ for $x \\in \\Q^\\times$, and to the [[adele ring|SEH68BA1QQ]].\nFor a general number field, the same method shows that every place comes from a real or complex embedding or from a nonzero prime ideal.\n",
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