{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4JF1SGYZ5Y"
    },
    {
      "field": "defines",
      "target": "ZYK5F0GBK6"
    }
  ],
  "body": "\n## Summary\n\nA discrete valuation is a [[valuation|B85T5VKR3M]] on a field whose values on nonzero elements are exactly the integers, see the [[definition|ZYK5F0GBK6]].\nIt counts the power of a fixed \"prime\" element in each nonzero element.\nThe [[p-adic valuation|03HDF50VN4]] on $\\Q$ and the order of vanishing of a rational function at a point are discrete valuations.\nThe elements of nonnegative value form a [[discrete valuation ring|MGM3CNJW9V]], one of the basic rings of number theory and algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nA discrete valuation gives each nonzero element an integer, its order, much as the exponent of $p$ in a fraction.\nPicture a rational function $f(x)$ near the point $0$: the valuation is the order of the zero of $f$ at $0$, or minus the order of the pole.\nFor example $x^3/(1 + x)$ has value $3$, and $1/x^2$ has value $-2$.\nThe values are whole numbers and every integer occurs, which is what \"discrete\" means here.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "12J20",
      "13F30"
    ],
    "name": "Discrete valuation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Valuation_(algebra)"
      }
    ],
    "specializes": [
      "B85T5VKR3M"
    ],
    "tag": "A2KKZAV29W"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/A2KKZAV29W/",
  "links": [
    {
      "field": "specializes",
      "target": "B85T5VKR3M"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/A2KKZAV29W.md",
  "msc": [
    "12J20",
    "13F30"
  ],
  "name": "Discrete valuation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A2K/A2KKZAV29W.md",
  "status": null,
  "tag": "A2KKZAV29W",
  "uri": "m2://A2KKZAV29W",
  "url": "https://math2.tamnd.com/wiki/discrete-valuation/",
  "writing": {
    "tag": "A2KKZAV29W",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}