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  "body": "\n## Summary\n\nA valuation on a [[field|04C5ZBG1WT]] measures how divisible an element is, with values in a totally ordered abelian group, see the [[definition|SVY882HKSJ]].\nThe value of a product is the sum of the values.\nThe value of a sum is at least the smaller of the two values.\nThe [[p-adic valuation|03HDF50VN4]] on the rationals is the model case.\nValuations give [[valuation rings|QSX12D8RQM]], [[non-Archimedean absolute values|1H5F1BP6XS]] and the whole theory of valued fields.\n\n## Explanations\n\n### L1 human\n\nA valuation asks how many times an element can be divided by a fixed \"small\" quantity.\nFor a prime $p$ and a nonzero rational number $x$, write $x = p^n a / b$ with $a$ and $b$ prime to $p$; the valuation of $x$ is $n$.\nSo $12 = 2^2 \\cdot 3$ has $2$-adic valuation $2$, and $3/8$ has $2$-adic valuation $-3$.\nMultiplying numbers adds their valuations, just as with exponents.\nAdding two numbers can only keep or raise the smaller valuation: $4 + 8 = 12$ has $2$-adic valuation $2$, the smaller of $2$ and $3$.\n",
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