{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]], and let $G = \\Aut(L/K)$ be the group of field automorphisms of $L$ that fix every element of $K$.\nLet $\\mathcal{S}$ be the set of subgroups of $G$ and let $\\mathcal{F}$ be the set of [[intermediate fields|02NBKVQFRQ]] of $L/K$, both ordered by inclusion.\nFor $H \\in \\mathcal{S}$, let $L^H = \\set{x \\in L : \\sigma(x) = x \\text{ for all } \\sigma \\in H}$ be the [[fixed field|0KMJ6919WS]] of $H$.\nFor $E \\in \\mathcal{F}$, let $\\Aut(L/E) = \\set{\\sigma \\in G : \\sigma(x) = x \\text{ for all } x \\in E}$.\nThe *Galois connection between subgroups and subfields* of $L/K$ is the pair of maps\n$$\\mathcal{S} \\to \\mathcal{F},\\ H \\mapsto L^H \\qquad \\text{and} \\qquad \\mathcal{F} \\to \\mathcal{S},\\ E \\mapsto \\Aut(L/E).$$\nIt is an antitone Galois connection: for all $H \\in \\mathcal{S}$ and $E \\in \\mathcal{F}$,\n$$E \\subseteq L^H \\iff H \\subseteq \\Aut(L/E).$$\n\n## Remarks\n\nBoth sides of the equivalence say that every element of $H$ fixes every element of $E$, which is the check.\nFrom the equivalence, $H \\subseteq \\Aut(L/L^H)$ and $E \\subseteq L^{\\Aut(L/E)}$, and each map reverses inclusions.\nExample: for $L = \\Q(\\sqrt{2}, \\sqrt{3})$ over $\\Q$, the group $G$ has four elements, and the maps are inverse bijections between its five subgroups and the five intermediate fields.\nNon-example of a bijection: for $L = \\Q(\\sqrt[3]{2})$ over $\\Q$, the group $G$ is trivial, because $\\sqrt[3]{2}$ is the only cube root of $2$ in the real field $L$.\nThen the map $E \\mapsto \\Aut(L/E)$ sends both $\\Q$ and $L$ to the trivial group, so it is not injective.\nThe maps are inverse bijections when $L/K$ is a finite [[Galois extension|0DS2F7GK05]], by the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "R3625YKQC0",
    "kind": "definition",
    "msc": [
      "12F10",
      "06A15"
    ],
    "name": "Galois connection between subgroups and subfields",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Galois_connection"
      }
    ],
    "tag": "FW68FVNKMJ",
    "uses": [
      "0WQ1A8SHTZ",
      "02NBKVQFRQ",
      "0KMJ6919WS"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/FW68FVNKMJ/",
  "links": [
    {
      "field": "defines",
      "target": "R3625YKQC0"
    },
    {
      "field": "uses",
      "target": "0WQ1A8SHTZ"
    },
    {
      "field": "uses",
      "target": "02NBKVQFRQ"
    },
    {
      "field": "uses",
      "target": "0KMJ6919WS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FW68FVNKMJ.md",
  "msc": [
    "12F10",
    "06A15"
  ],
  "name": "Galois connection between subgroups and subfields",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FW6/FW68FVNKMJ.md",
  "status": null,
  "tag": "FW68FVNKMJ",
  "uri": "m2://FW68FVNKMJ",
  "url": "https://math2.tamnd.com/wiki/galois-connection-between-subgroups-and-subfields-definition/",
  "writing": {
    "tag": "FW68FVNKMJ",
    "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
      }
    ]
  }
}