{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "SWDN96J72T"
    }
  ],
  "body": "\n## Definition\n\nLet $\\Omega$ be a field and let $E$ and $F$ be [[subfields|07QY5RPFBR]] of $\\Omega$.\nThe *compositum* $EF$ of $E$ and $F$ is the intersection of all subfields of $\\Omega$ that contain $E \\cup F$.\nMore generally, the compositum of a family of subfields of $\\Omega$ is the intersection of all subfields of $\\Omega$ containing every member.\n\n## Remarks\n\nThe compositum $EF$ is the smallest subfield of $\\Omega$ containing $E$ and $F$, so $EF = E(F) = F(E)$ in the notation of [[fields generated by adjoining elements|0MA5B744K3]].\nIts elements are the quotients $a/b$ with $b \\ne 0$, where $a$ and $b$ are finite sums of products $xy$ with $x \\in E$ and $y \\in F$.\nIf $K \\subseteq E \\cap F$ and $E/K$ is finite, then $[EF:F] \\le [E:K]$.\n\nExample: in $\\Omega = \\C$, the compositum of $E = \\Q(\\sqrt{2})$ and $F = \\Q(\\sqrt{3})$ is $\\Q(\\sqrt{2}, \\sqrt{3})$.\nCheck: this field contains $E$ and $F$, and any subfield of $\\C$ containing $\\sqrt{2}$ and $\\sqrt{3}$ contains $\\Q(\\sqrt{2}, \\sqrt{3})$.\n\nNon-example: the union $E \\cup F$ is not the compositum, because it is not a field.\nIndeed $\\sqrt{2} + \\sqrt{3}$ lies in neither $E$ nor $F$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "T1S29NN8PE",
    "kind": "definition",
    "msc": [
      "12F05",
      "12F99"
    ],
    "name": "Compositum of fields",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tensor_product_of_fields"
      }
    ],
    "tag": "X4ZZKC319Q",
    "uses": [
      "07QY5RPFBR",
      "0MA5B744K3"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/X4ZZKC319Q/",
  "links": [
    {
      "field": "defines",
      "target": "T1S29NN8PE"
    },
    {
      "field": "uses",
      "target": "07QY5RPFBR"
    },
    {
      "field": "uses",
      "target": "0MA5B744K3"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/X4ZZKC319Q.md",
  "msc": [
    "12F05",
    "12F99"
  ],
  "name": "Compositum of fields",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X4Z/X4ZZKC319Q.md",
  "status": null,
  "tag": "X4ZZKC319Q",
  "uri": "m2://X4ZZKC319Q",
  "url": "https://math2.tamnd.com/wiki/compositum-of-fields-definition/",
  "writing": {
    "tag": "X4ZZKC319Q",
    "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
      }
    ]
  }
}