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  "body": "\n## Summary\n\nA discrete valuation ring is a [[principal ideal domain|0CMD0BCH8T]] with exactly one nonzero [[prime ideal|00MHHZ9SNE]].\nIt is the ring of elements of nonnegative value for a [[discrete valuation|A2KKZAV29W]] on its field of fractions.\nDiscrete valuation rings are among the simplest rings that are not fields: their ideals form a single chain of powers of one element.\nThey describe the local behaviour of curves at smooth points and of number rings at a prime.\n\n## Explanations\n\n### L1 human\n\nIn a discrete valuation ring there is one special element $\\pi$, a uniformizer, and every nonzero element is a unit times a power of $\\pi$.\nThe power of $\\pi$ measures \"how divisible\" an element is.\nThe main example is the ring $\\Z_{(p)}$ of fractions $a/b$ with $p \\nmid b$, where $\\pi = p$.\nFor instance, $12/5 = 3/5 \\cdot 2^2$ in $\\Z_{(2)}$, so its value is $2$.\nAnother example is the ring of formal power series $k[\\![t]\\!]$, where $\\pi = t$ and the value of a series is its lowest degree.\n",
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