{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "AVCZ6HVBA7"
    },
    {
      "field": "specializes",
      "target": "G5PVJMM37Z"
    },
    {
      "field": "defines",
      "target": "HMF8G4EBDX"
    }
  ],
  "body": "\n## Summary\n\nA transcendental extension is a [[field extension|0EQWVHF3W5]] that is not [[algebraic|0PTBM7K0R2]].\nSo it contains at least one [[transcendental element|0NQMZ89X4A]], an element that is a root of no nonzero polynomial over the base field.\nThe size of such an extension is measured by its [[transcendence degree|D7TR0YWR9H]], the number of independent transcendental elements it needs.\nFields of rational functions and function fields of varieties are the main examples.\n\n## Explanations\n\n### L1 human\n\nAn algebraic extension adds roots of polynomials, such as $\\sqrt 2$ added to $\\Q$.\nA transcendental extension adds something that solves no polynomial equation, such as $\\pi$ added to $\\Q$, or a free variable $t$.\nThe field $\\Q(t)$ of rational functions in $t$ is the simplest example: no nonzero polynomial with rational coefficients vanishes at $t$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "kind": "concept",
    "msc": [
      "12F20"
    ],
    "name": "Transcendental extension",
    "same_as": {
      "wikidata": "Q3733413",
      "wikipedia": "Transcendental extension"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Transcendental_extension"
      }
    ],
    "specializes": [
      "0EQWVHF3W5"
    ],
    "tag": "12JQCGTQ8W"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/12JQCGTQ8W/",
  "links": [
    {
      "field": "specializes",
      "target": "0EQWVHF3W5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/12JQCGTQ8W.md",
  "msc": [
    "12F20"
  ],
  "name": "Transcendental extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/12J/12JQCGTQ8W.md",
  "status": null,
  "tag": "12JQCGTQ8W",
  "uri": "m2://12JQCGTQ8W",
  "url": "https://math2.tamnd.com/wiki/transcendental-extension/",
  "writing": {
    "tag": "12JQCGTQ8W",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}