{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nA *separating transcendence basis* of $L/K$ is a [[transcendence basis|E45PA267S4]] $B$ of $L/K$ such that $L$ is a [[separable|0N019F6RE9]] algebraic extension of $K(B)$.\nThe extension $L/K$ is *separably generated* if it has a separating transcendence basis.\n\n## Remarks\n\nAn algebraic extension is separably generated if and only if it is separable, since then $B = \\emptyset$.\nIn characteristic $0$ every extension is separably generated.\nEvery finitely generated extension of a [[perfect field|VWBVKT794W]] is separably generated.\nExample: $K(t)/K$ is separably generated, with separating transcendence basis $\\set{t}$.\nNon-example: let $K = \\F_p(t)$ and $L = K(t^{1/p})$; then $L/K$ is algebraic and [[purely inseparable|966GNVGB8A]] of degree $p$, so its only transcendence basis is $\\emptyset$ and $L/K$ is not separable.\nNot every transcendence basis of a separably generated extension is separating: $\\set{t^p}$ is a transcendence basis of $\\F_p(t)/\\F_p$, but $\\F_p(t)/\\F_p(t^p)$ is purely inseparable.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "1Z2BCP64SZ",
    "kind": "definition",
    "msc": [
      "12F20",
      "12F15"
    ],
    "name": "Separably generated extension",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Separable_extension"
      }
    ],
    "tag": "QCBYAWBYWM",
    "uses": [
      "0WQ1A8SHTZ",
      "E45PA267S4",
      "0N019F6RE9",
      "VWBVKT794W",
      "966GNVGB8A"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/QCBYAWBYWM/",
  "links": [
    {
      "field": "defines",
      "target": "1Z2BCP64SZ"
    },
    {
      "field": "uses",
      "target": "0WQ1A8SHTZ"
    },
    {
      "field": "uses",
      "target": "E45PA267S4"
    },
    {
      "field": "uses",
      "target": "0N019F6RE9"
    },
    {
      "field": "uses",
      "target": "VWBVKT794W"
    },
    {
      "field": "uses",
      "target": "966GNVGB8A"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QCBYAWBYWM.md",
  "msc": [
    "12F20",
    "12F15"
  ],
  "name": "Separably generated extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QCB/QCBYAWBYWM.md",
  "status": null,
  "tag": "QCBYAWBYWM",
  "uri": "m2://QCBYAWBYWM",
  "url": "https://math2.tamnd.com/wiki/separably-generated-extension-definition/",
  "writing": {
    "tag": "QCBYAWBYWM",
    "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
      }
    ]
  }
}