{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $K$ be a field extension of $k$ that is [[finitely generated|AC5TFFQT9M]] as a $k$-algebra.\nThen $K$ is a [[finite extension|0C1Y04SZ1W]] of $k$.\n\n## Remarks\n\nBy [[Noether normalization|7Z28JY8QW4]], $K$ is finite over a polynomial ring $k[y_1, \\ldots, y_d]$.\nBy [[integral extensions of fields|5GCANV7337]], $k[y_1, \\ldots, y_d]$ is then a field, which forces $d = 0$, since $y_1$ has no inverse in a polynomial ring when $d \\ge 1$.\nSo $K$ is finite over $k$.\nAnother proof uses the [[Artin–Tate lemma|4MBR0W0S6N]].\n\nExample: $\\Q(\\sqrt{2}) = \\Q[\\sqrt{2}]$ is a field generated by one element as a $\\Q$-algebra, and it is finite over $\\Q$ of degree $2$.\n\nThe lemma is about generation as an algebra.\nThe field $k(t)$ of rational functions is generated by $t$ as a field, but it is not a finitely generated $k$-algebra, and it is not finite over $k$.\n\n## Explanations\n\n### L1 human\n\nA field that can be built from a smaller field with finitely many elements, using only addition and multiplication, is a finite extension of it.\nInverses come for free in such a field, so no transcendental element can occur.\n\n### L2 human\n\nZariski's lemma is the algebraic core of the [[weak Nullstellensatz|ZWS0YWGF17]].\nIf $\\mathfrak{m}$ is a maximal ideal of $k[x_1, \\ldots, x_n]$, then $k[x_1, \\ldots, x_n]/\\mathfrak{m}$ is a field finitely generated as a $k$-algebra, so it is finite over $k$.\nWhen $k$ is algebraically closed, this residue field is $k$ itself, and $\\mathfrak{m}$ is the ideal of a point.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "AC5TFFQT9M",
      "0C1Y04SZ1W"
    ],
    "kind": "statement",
    "msc": [
      "13E15",
      "12F05"
    ],
    "name": "Zariski's lemma",
    "role": "lemma",
    "same_as": {
      "wikidata": "Q8066794",
      "wikipedia": "Zariski's lemma"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Zariski's_lemma"
      }
    ],
    "tag": "YBJ2VQHFDQ",
    "uses": [
      "04C5ZBG1WT",
      "AC5TFFQT9M",
      "0C1Y04SZ1W"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/YBJ2VQHFDQ/",
  "links": [
    {
      "field": "about",
      "target": "AC5TFFQT9M"
    },
    {
      "field": "about",
      "target": "0C1Y04SZ1W"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    },
    {
      "field": "uses",
      "target": "AC5TFFQT9M"
    },
    {
      "field": "uses",
      "target": "0C1Y04SZ1W"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YBJ2VQHFDQ.md",
  "msc": [
    "13E15",
    "12F05"
  ],
  "name": "Zariski's lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YBJ/YBJ2VQHFDQ.md",
  "status": null,
  "tag": "YBJ2VQHFDQ",
  "uri": "m2://YBJ2VQHFDQ",
  "url": "https://math2.tamnd.com/wiki/zariskis-lemma/",
  "writing": {
    "tag": "YBJ2VQHFDQ",
    "kind": "statement",
    "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
      }
    ]
  }
}