{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "02NZZ1J0FH"
    },
    {
      "field": "uses",
      "target": "0P6SBR7YM2"
    }
  ],
  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $S \\subseteq L$.\nThe field $K(S)$ is the intersection of all the [[subfields|07QY5RPFBR]] of $L$ that contain $K$ and $S$.\nIt is the smallest [[intermediate field|02NBKVQFRQ]] of $L/K$ that contains $S$.\n\n## Remarks\n\nFor $S = \\{a_1, \\dots, a_n\\}$ one writes $K(a_1, \\dots, a_n)$.\nThe elements of $K(S)$ are the quotients $f(s_1, \\dots, s_n)/g(s_1, \\dots, s_n)$ with $f, g$ polynomials over $K$, $s_i \\in S$ and $g(s_1, \\dots, s_n) \\neq 0$.\nOne has $K(a, b) = K(a)(b)$.\nFor example, $\\mathbb{Q}(\\sqrt{2}) = \\{ p + q \\sqrt{2} : p, q \\in \\mathbb{Q} \\}$ inside $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nTo adjoin elements to a field, you take the smallest field that contains the old field and the new elements.\nIt contains everything you can get from them by addition, subtraction, multiplication and division.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "05NAS7AGMZ",
    "kind": "definition",
    "msc": [
      "12F05"
    ],
    "name": "Field generated by adjoining elements",
    "sources": [
      {
        "locator": "Extension field",
        "url": "https://en.wikipedia.org/wiki/Field_extension"
      }
    ],
    "tag": "0MA5B744K3",
    "uses": [
      "0WQ1A8SHTZ",
      "07QY5RPFBR",
      "02NBKVQFRQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0MA5B744K3/",
  "links": [
    {
      "field": "defines",
      "target": "05NAS7AGMZ"
    },
    {
      "field": "uses",
      "target": "0WQ1A8SHTZ"
    },
    {
      "field": "uses",
      "target": "07QY5RPFBR"
    },
    {
      "field": "uses",
      "target": "02NBKVQFRQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0MA5B744K3.md",
  "msc": [
    "12F05"
  ],
  "name": "Field generated by adjoining elements",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0MA/0MA5B744K3.md",
  "status": null,
  "tag": "0MA5B744K3",
  "uri": "m2://0MA5B744K3",
  "url": "https://math2.tamnd.com/wiki/field-generated-by-adjoining-elements-definition/",
  "writing": {
    "tag": "0MA5B744K3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}