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  "body": "\n## Summary\n\nFor a prime $p$, the $p$-adic absolute value of a nonzero rational number $x$ is $\\abs{x}_p = p^{-v_p(x)}$, where $v_p$ is the [[p-adic valuation|03HDF50VN4]].\nIt is a [[non-Archimedean absolute value|1H5F1BP6XS]] on $\\Q$.\nCompleting $\\Q$ for it gives the field of [[p-adic numbers|BT6T702C28]].\nTogether with the usual [[absolute value|0FFXKTKKQC]], these are all the nontrivial absolute values on $\\Q$ up to equivalence.\n\n## Explanations\n\n### L1 human\n\nThe $p$-adic absolute value measures how divisible a number is by $p$.\nThe more factors of $p$ a number has, the smaller its $p$-adic size.\nFor example, $\\abs{12}_2 = 1/4$ because $12 = 2^2 \\cdot 3$, and $\\abs{3/8}_2 = 8$ because $8 = 2^3$ is in the denominator.\nNumbers not divisible by $p$, in numerator or denominator, have size $1$.\n",
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