{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nThe *transcendence degree* of $L/K$, written $\\operatorname{trdeg}_K L$, is the [[cardinality|0JTGRV4WT4]] of a [[transcendence basis|E45PA267S4]] of $L/K$.\n\n## Remarks\n\nThe definition makes sense because [[a transcendence basis exists|W635VBXEXG]] and any two transcendence bases have the same cardinality; the proof uses an exchange argument like the [[Steinitz exchange lemma|08ZDRN6H41]].\nExample: $\\operatorname{trdeg}_K K(x_1, \\dots, x_n) = n$, with transcendence basis $\\set{x_1, \\dots, x_n}$.\nExample: $\\operatorname{trdeg}_\\Q \\R$ is the cardinality of the continuum, since $\\R$ is uncountable and the algebraic closure of a countable field is countable.\nNon-example: for $L = \\Q(\\pi, \\pi^2)$, the transcendence degree over $\\Q$ is $1$, not $2$, because $\\set{\\pi, \\pi^2}$ is dependent and $\\set{\\pi}$ is a transcendence basis.\nIt is additive in towers: $\\operatorname{trdeg}_K M = \\operatorname{trdeg}_L M + \\operatorname{trdeg}_K L$ for $K \\subseteq L \\subseteq M$.\nThe transcendence degree is $0$ if and only if the extension is [[algebraic|09EYMKMSB9]].\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "D7TR0YWR9H",
    "kind": "definition",
    "msc": [
      "12F20"
    ],
    "name": "Transcendence degree",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Transcendental_extension"
      }
    ],
    "tag": "NVQ0N7QTCR",
    "uses": [
      "0WQ1A8SHTZ",
      "E45PA267S4",
      "W635VBXEXG",
      "0JTGRV4WT4",
      "09EYMKMSB9",
      "08ZDRN6H41"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/NVQ0N7QTCR/",
  "links": [
    {
      "field": "defines",
      "target": "D7TR0YWR9H"
    },
    {
      "field": "uses",
      "target": "0WQ1A8SHTZ"
    },
    {
      "field": "uses",
      "target": "E45PA267S4"
    },
    {
      "field": "uses",
      "target": "W635VBXEXG"
    },
    {
      "field": "uses",
      "target": "0JTGRV4WT4"
    },
    {
      "field": "uses",
      "target": "09EYMKMSB9"
    },
    {
      "field": "uses",
      "target": "08ZDRN6H41"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NVQ0N7QTCR.md",
  "msc": [
    "12F20"
  ],
  "name": "Transcendence degree",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NVQ/NVQ0N7QTCR.md",
  "status": null,
  "tag": "NVQ0N7QTCR",
  "uri": "m2://NVQ0N7QTCR",
  "url": "https://math2.tamnd.com/wiki/transcendence-degree-definition/",
  "writing": {
    "tag": "NVQ0N7QTCR",
    "kind": "definition",
    "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
      }
    ]
  }
}