{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $B$ be a commutative ring and let $A$ be a [[subring|0GNCRPYXYV]] of $B$.\nThe *integral closure of $A$ in $B$* is the set\n$$\\overline{A} = \\set{b \\in B : b \\text{ is integral over } A}$$\nof the elements of $B$ that are [[integral|1EHN4KPSFS]] over $A$.\nThe ring $A$ is *integrally closed in $B$* if $\\overline{A} = A$.\nFor an [[integral domain|0QVD7WBFGN]] $A$, the integral closure of $A$ in its [[field of fractions|0F2PTGX1A2]] is the *normalization* of $A$.\n\n## Remarks\n\nThe set $\\overline{A}$ is a subring of $B$ that contains $A$, and it is integrally closed in $B$ by [[transitivity of integrality|ZXZMQM818D]].\n\nExample: the integral closure of $\\Z$ in $\\Q(i)$ is $\\Z[i]$.\nIf $a + bi$ with $a, b \\in \\Q$ is integral, then so is its conjugate, so $2a$ and $a^2 + b^2$ are rational and integral over $\\Z$, hence integers, and this forces $a, b \\in \\Z$.\n\nNon-example: $\\Z[\\sqrt{5}]$ is not integrally closed in $\\Q(\\sqrt{5})$.\nThe element $(1 + \\sqrt{5})/2$ is a root of $x^2 - x - 1$, so it is integral over $\\Z$, but it is not in $\\Z[\\sqrt{5}]$.\nThe integral closure of $\\Z$ in $\\Q(\\sqrt{5})$ is $\\Z[(1 + \\sqrt{5})/2]$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "G0V86SJBQW",
    "kind": "definition",
    "msc": [
      "13B22",
      "13B21"
    ],
    "name": "Integral closure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Integral_element"
      }
    ],
    "tag": "DRJPG78875",
    "uses": [
      "1EHN4KPSFS",
      "0GNCRPYXYV",
      "0QVD7WBFGN",
      "0F2PTGX1A2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/DRJPG78875/",
  "links": [
    {
      "field": "defines",
      "target": "G0V86SJBQW"
    },
    {
      "field": "uses",
      "target": "1EHN4KPSFS"
    },
    {
      "field": "uses",
      "target": "0GNCRPYXYV"
    },
    {
      "field": "uses",
      "target": "0QVD7WBFGN"
    },
    {
      "field": "uses",
      "target": "0F2PTGX1A2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/DRJPG78875.md",
  "msc": [
    "13B22",
    "13B21"
  ],
  "name": "Integral closure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DRJ/DRJPG78875.md",
  "status": null,
  "tag": "DRJPG78875",
  "uri": "m2://DRJPG78875",
  "url": "https://math2.tamnd.com/wiki/integral-closure-definition/",
  "writing": {
    "tag": "DRJPG78875",
    "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
      }
    ]
  }
}