{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "CAEHMSX9RT"
    },
    {
      "field": "uses",
      "target": "DQ564JNFYE"
    },
    {
      "field": "uses",
      "target": "NVQ0N7QTCR"
    },
    {
      "field": "uses",
      "target": "QCBYAWBYWM"
    },
    {
      "field": "uses",
      "target": "W635VBXEXG"
    },
    {
      "field": "defines",
      "target": "CYCJ46W6P6"
    },
    {
      "field": "about",
      "target": "W635VBXEXG"
    }
  ],
  "body": "\n## Summary\n\nA transcendence basis of a [[field extension|0EQWVHF3W5]] $L/K$ is a set $B$ of [[algebraically independent|KV4PT6KW10]] elements such that $L$ is [[algebraic|0PTBM7K0R2]] over $K(B)$.\nIt plays the role that a [[basis|0KCQ9Y3QKA]] plays for a vector space, with polynomial relations in place of linear ones.\n[[Every extension has one|W635VBXEXG]], and all of them have the same cardinality, the [[transcendence degree|D7TR0YWR9H]].\n\n## Explanations\n\n### L1 human\n\nA transcendence basis is a largest set of free variables inside a field extension.\nEverything else in the big field is tied to these variables by polynomial equations.\nIn $\\Q(x, \\sqrt{x^3 + 1})$, the single element $x$ is a transcendence basis over $\\Q$, since $y = \\sqrt{x^3 + 1}$ satisfies $y^2 = x^3 + 1$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "12F20"
    ],
    "name": "Transcendence basis",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Transcendental_extension"
      }
    ],
    "tag": "E45PA267S4"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/E45PA267S4/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/E45PA267S4.md",
  "msc": [
    "12F20"
  ],
  "name": "Transcendence basis",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E45/E45PA267S4.md",
  "status": null,
  "tag": "E45PA267S4",
  "uri": "m2://E45PA267S4",
  "url": "https://math2.tamnd.com/wiki/transcendence-basis/",
  "writing": {
    "tag": "E45PA267S4",
    "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
      }
    ]
  }
}