{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2A7CYKQ2XW"
    },
    {
      "field": "uses",
      "target": "HCD6W2XK6W"
    },
    {
      "field": "uses",
      "target": "J8YYYP5CDN"
    },
    {
      "field": "uses",
      "target": "XDQYMY78TS"
    },
    {
      "field": "defines",
      "target": "DRJPG78875"
    },
    {
      "field": "about",
      "target": "HCD6W2XK6W"
    },
    {
      "field": "about",
      "target": "J8YYYP5CDN"
    }
  ],
  "body": "\n## Summary\n\nThe integral closure of a ring $A$ in a larger commutative ring $B$ is the set of elements of $B$ that are [[integral|1EHN4KPSFS]] over $A$.\nIt is a subring of $B$ that contains $A$, and it is the largest subring of $B$ that is [[integral over|B7X0KFY1S1]] $A$.\nFor an integral domain $A$, its integral closure in its own field of fractions is called its normalization.\nThe ring of integers of a number field is the integral closure of $\\Z$ in that field.\n\n## Explanations\n\n### L1 human\n\nTake all elements of the larger ring that satisfy a monic equation with coefficients in $A$; together they form a ring.\nInside $\\Q(\\sqrt{5})$, the integral closure of $\\Z$ contains $\\sqrt{5}$ but also the golden ratio $(1 + \\sqrt{5})/2$, a root of $x^2 - x - 1$.\nSo the \"integers\" of $\\Q(\\sqrt{5})$ form the ring $\\Z[(1 + \\sqrt{5})/2]$, which is larger than $\\Z[\\sqrt{5}]$.\nPassing to the integral closure repairs rings that miss some of their natural elements.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13B22",
      "13B21"
    ],
    "name": "Integral closure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Integral_element"
      }
    ],
    "tag": "G0V86SJBQW"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/G0V86SJBQW/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/G0V86SJBQW.md",
  "msc": [
    "13B22",
    "13B21"
  ],
  "name": "Integral closure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G0V/G0V86SJBQW.md",
  "status": null,
  "tag": "G0V86SJBQW",
  "uri": "m2://G0V86SJBQW",
  "url": "https://math2.tamnd.com/wiki/integral-closure/",
  "writing": {
    "tag": "G0V86SJBQW",
    "kind": "concept",
    "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
      }
    ]
  }
}