{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "D186VNHFZJ"
    }
  ],
  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $\\Gamma$ be a totally ordered [[abelian group|071DDFVG4J]], written additively.\nAdd to $\\Gamma$ a symbol $\\infty$ with $\\gamma \u003c \\infty$ and $\\gamma + \\infty = \\infty + \\gamma = \\infty + \\infty = \\infty$ for all $\\gamma \\in \\Gamma$.\nA *valuation* on $K$ with values in $\\Gamma$ is a map $v \\colon K \\to \\Gamma \\cup \\set{\\infty}$ such that:\n\n1. $v(x) = \\infty$ if and only if $x = 0$.\n2. $v(xy) = v(x) + v(y)$ for all $x, y \\in K$.\n3. $v(x + y) \\ge \\min(v(x), v(y))$ for all $x, y \\in K$.\n\nThe subgroup $v(K^\\times)$ of $\\Gamma$ is the *value group* of $v$.\nThe pair $(K, v)$ is a *valued field*.\n\n## Remarks\n\nExample: for a prime $p$, the [[p-adic valuation|04VCPS30ZE]] $v_p$ on $\\Q$ with values in $\\Z$ is a valuation.\nIf $x = p^m a/b$ and $y = p^n c/d$ with $a, b, c, d$ prime to $p$ and $m \\le n$, then $xy = p^{m+n} ac/(bd)$ gives clause 2, and $x + y = p^m (ad + p^{n-m} cb)/(bd)$ gives $v_p(x + y) \\ge m$, which is clause 3.\nExample: on the field $k(t)$ of rational functions over a field $k$, the map $f/g \\mapsto \\deg g - \\deg f$ is a valuation with values in $\\Z$.\nNon-example: the map $v \\colon \\Q \\to \\Z \\cup \\set{\\infty}$ with $v(0) = \\infty$ and $v(x) = 1$ for $x \\ne 0$ fails clause 2, since $v(1 \\cdot 1) = 1 \\ne 2 = v(1) + v(1)$.\nEvery field has the *trivial valuation*, with $v(x) = 0$ for all $x \\ne 0$.\nClauses 1 and 2 give $v(1) = 0$, $v(-x) = v(x)$ and $v(x^{-1}) = -v(x)$ for $x \\ne 0$.\nThe set $\\set{x \\in K : v(x) \\ge 0}$ is a [[valuation ring|QSX12D8RQM]] of $K$.\nSome texts write valuations multiplicatively, with the reverse order; for $\\Gamma \\subseteq \\R$ the map $x \\mapsto c^{v(x)}$ with $0 \u003c c \u003c 1$ is a [[non-Archimedean absolute value|1H5F1BP6XS]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "B85T5VKR3M",
    "kind": "definition",
    "msc": [
      "12J20",
      "13F30"
    ],
    "name": "Valuation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Valuation_(algebra)"
      }
    ],
    "tag": "SVY882HKSJ",
    "uses": [
      "0QA8CHH8MG",
      "071DDFVG4J"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/SVY882HKSJ/",
  "links": [
    {
      "field": "defines",
      "target": "B85T5VKR3M"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "071DDFVG4J"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SVY882HKSJ.md",
  "msc": [
    "12J20",
    "13F30"
  ],
  "name": "Valuation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SVY/SVY882HKSJ.md",
  "status": null,
  "tag": "SVY882HKSJ",
  "uri": "m2://SVY882HKSJ",
  "url": "https://math2.tamnd.com/wiki/valuation-definition/",
  "writing": {
    "tag": "SVY882HKSJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}