{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "GTGFXW37YM"
    },
    {
      "field": "uses",
      "target": "KHTCR3VFST"
    },
    {
      "field": "uses",
      "target": "TDCGM4WR88"
    }
  ],
  "body": "\n## Summary\n\nThe normal closure of an algebraic extension $L/K$ is the smallest extension of $L$ that is [[normal|08S0NW8SES]] over $K$.\nIt is obtained by adjoining to $L$ all the roots of the [[minimal polynomials|0VX9EP2GS8]] over $K$ of the elements of $L$.\nIt exists and is unique up to an isomorphism over $L$.\nWhen $L/K$ is finite and separable, its normal closure is the smallest [[Galois extension|0QMZAQ6QQ4]] of $K$ containing $L$.\nThis lets Galois theory study extensions that are not themselves Galois.\n\n## Explanations\n\n### L1 human\n\nSome extensions contain one root of a polynomial but not the others.\nThe normal closure adds the missing roots, and nothing more.\nFor example, $\\Q(\\sqrt[3]{2})$ contains the real cube root of $2$ but not the two complex cube roots.\nAdding them gives $\\Q(\\sqrt[3]{2}, \\omega)$ with $\\omega = e^{2\\pi i/3}$, a field of degree $6$ over $\\Q$.\nThis field is the normal closure of $\\Q(\\sqrt[3]{2})$ over $\\Q$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "12F05",
      "12F10"
    ],
    "name": "Normal closure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Normal_extension"
      }
    ],
    "tag": "6XM4CCVTKE"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/6XM4CCVTKE/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/6XM4CCVTKE.md",
  "msc": [
    "12F05",
    "12F10"
  ],
  "name": "Normal closure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6XM/6XM4CCVTKE.md",
  "status": null,
  "tag": "6XM4CCVTKE",
  "uri": "m2://6XM4CCVTKE",
  "url": "https://math2.tamnd.com/wiki/normal-closure/",
  "writing": {
    "tag": "6XM4CCVTKE",
    "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
      }
    ]
  }
}