{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "Q2XBHJ96GV"
    }
  ],
  "body": "\n## Statement\n\nLet $A \\subseteq B$ be [[integral domains|0QVD7WBFGN]] with $B$ [[integral over|B7X0KFY1S1]] $A$.\nThen $A$ is a [[field|04C5ZBG1WT]] if and only if $B$ is a field.\n\n## Remarks\n\nSuppose $A$ is a field and let $b \\in B$ be nonzero.\nTake a monic relation $b^n + a_{n-1} b^{n-1} + \\cdots + a_0 = 0$ of least degree; since $B$ is a domain, $a_0 \\ne 0$.\nThen $b^{-1} = -a_0^{-1}(b^{n-1} + a_{n-1} b^{n-2} + \\cdots + a_1)$ lies in $B$.\nConversely, suppose $B$ is a field and let $a \\in A$ be nonzero.\nThen $a^{-1} \\in B$ is integral over $A$, say $a^{-m} + c_{m-1} a^{-m+1} + \\cdots + c_0 = 0$, and multiplying by $a^{m-1}$ gives $a^{-1} = -(c_{m-1} + c_{m-2} a + \\cdots + c_0 a^{m-1}) \\in A$.\n\nExample: $\\Q \\subseteq \\Q(\\sqrt{2})$ is integral and both are fields; $\\Z \\subseteq \\Z[i]$ is integral and neither is a field.\n\nThe hypothesis that $B$ is a domain is needed for \"$A$ field implies $B$ field\": the ring $k[x]/(x^2)$ is integral over the field $k$, but it is not a field.\n\n## Explanations\n\n### L1 human\n\nIn an integral extension of domains, the small ring is a field exactly when the big ring is.\nSo integral extensions cannot turn a non-field into a field, nor the reverse.\n\n### L2 human\n\nThis is the step that links maximal ideals across an integral extension: for a prime $\\mathfrak{q}$ of $B$, apply it to $A/(\\mathfrak{q} \\cap A) \\subseteq B/\\mathfrak{q}$.\nIt gives that $\\mathfrak{q}$ is maximal if and only if $\\mathfrak{q} \\cap A$ is maximal, and it is used in the proofs of [[lying over|YZ97GKYY3P]], [[incomparability|Q2XBHJ96GV]] and [[Zariski's lemma|YBJ2VQHFDQ]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "B7X0KFY1S1"
    ],
    "kind": "statement",
    "msc": [
      "13B21"
    ],
    "name": "Integral extensions of fields",
    "role": "proposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Integral_element"
      }
    ],
    "tag": "5GCANV7337",
    "uses": [
      "B7X0KFY1S1",
      "0QVD7WBFGN",
      "04C5ZBG1WT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/5GCANV7337/",
  "links": [
    {
      "field": "about",
      "target": "B7X0KFY1S1"
    },
    {
      "field": "uses",
      "target": "B7X0KFY1S1"
    },
    {
      "field": "uses",
      "target": "0QVD7WBFGN"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/5GCANV7337.md",
  "msc": [
    "13B21"
  ],
  "name": "Integral extensions of fields",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5GC/5GCANV7337.md",
  "status": null,
  "tag": "5GCANV7337",
  "uri": "m2://5GCANV7337",
  "url": "https://math2.tamnd.com/wiki/integral-extensions-of-fields/",
  "writing": {
    "tag": "5GCANV7337",
    "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
      }
    ]
  }
}