{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "BMGE1ZAFQM"
    },
    {
      "field": "specializes",
      "target": "RVR5QFFD77"
    }
  ],
  "body": "\n## Summary\n\nGalois theory studies a field extension through its group of automorphisms.\nIts central result, the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], matches the intermediate fields of a finite [[Galois extension|0DS2F7GK05]] with the subgroups of its [[Galois group|0XVR72M51M]].\nIt explains which polynomial equations are [[solvable by radicals|J7YS0DA3HQ]] and which constructions with straightedge and compass are possible.\n\n## Explanations\n\n### L1 human\n\nThe roots of a polynomial can often be permuted without breaking any algebraic relation among them.\nThese permutations form a group, and Galois theory reads properties of the equation off this group.\nFor $x^2 - 2$, the two roots $\\sqrt{2}$ and $-\\sqrt{2}$ can be exchanged, and the group has two elements.\nOver a field of characteristic $0$, a polynomial can be solved with roots and the four basic operations exactly when its group is [[solvable|0P3KZK0FTV]].\nThis is why there is a formula for degree $4$ and none for the general equation of degree $5$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "12F10"
    ],
    "name": "Galois theory",
    "same_as": {
      "wikidata": "Q92552",
      "wikipedia": "Galois theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Galois_theory"
      }
    ],
    "tag": "WNQDKQYZ2R"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/WNQDKQYZ2R/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/WNQDKQYZ2R.md",
  "msc": [
    "12F10"
  ],
  "name": "Galois theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WNQ/WNQDKQYZ2R.md",
  "status": null,
  "tag": "WNQDKQYZ2R",
  "uri": "m2://WNQDKQYZ2R",
  "url": "https://math2.tamnd.com/wiki/galois-theory/",
  "writing": {
    "tag": "WNQDKQYZ2R",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}