{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "BMGE1ZAFQM"
    },
    {
      "field": "defines",
      "target": "FW68FVNKMJ"
    }
  ],
  "body": "\n## Summary\n\nLet $L/K$ be a field extension.\nOne map sends a group $H$ of automorphisms of $L$ over $K$ to its fixed field.\nThe other sends an intermediate field $E$ to the group of automorphisms that fix $E$.\nTogether they form an antitone Galois connection.\nBoth maps reverse inclusions, and applying one after the other can only enlarge a subgroup or an intermediate field.\nFor a finite [[Galois extension|0DS2F7GK05]] the two maps are inverse bijections; this is the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].\n\n## Explanations\n\n### L1 human\n\nThink of two lists side by side: the subgroups of the automorphism group, and the fields between $K$ and $L$.\nEach subgroup points to the field of elements it leaves fixed, and each field points to the group of automorphisms that leave it fixed.\nA bigger group fixes fewer elements, so the arrows turn the order upside down.\nFor $L = \\Q(\\sqrt{2})$ over $K = \\Q$, the trivial group points to $L$, and the group generated by $\\sqrt{2} \\mapsto -\\sqrt{2}$ points to $\\Q$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "12F10",
      "06A15"
    ],
    "name": "Galois connection between subgroups and subfields",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Galois_connection"
      }
    ],
    "tag": "R3625YKQC0"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/R3625YKQC0/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/R3625YKQC0.md",
  "msc": [
    "12F10",
    "06A15"
  ],
  "name": "Galois connection between subgroups and subfields",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R36/R3625YKQC0.md",
  "status": null,
  "tag": "R3625YKQC0",
  "uri": "m2://R3625YKQC0",
  "url": "https://math2.tamnd.com/wiki/galois-connection-between-subgroups-and-subfields/",
  "writing": {
    "tag": "R3625YKQC0",
    "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
      }
    ]
  }
}