{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2A7CYKQ2XW"
    },
    {
      "field": "uses",
      "target": "4K6F157663"
    },
    {
      "field": "uses",
      "target": "DRJPG78875"
    },
    {
      "field": "uses",
      "target": "ZXZMQM818D"
    },
    {
      "field": "defines",
      "target": "X83RX7KDSP"
    }
  ],
  "body": "\n## Summary\n\nAn element $b$ of a [[commutative ring|0310REJJY8]] $B$ is integral over a subring $A$ if it is a root of a [[monic polynomial|0C3QBCR46M]] over $A$.\nIt extends to rings the notion of an [[algebraic element|0H628RSGWX]] of a field extension, with the extra demand that the leading coefficient is $1$.\nOver $\\Z$, the integral elements of $\\C$ are the algebraic integers, the basic objects of algebraic number theory.\nIntegral elements are the starting point of the theory of [[integral extensions|B7X0KFY1S1]], integral closure and normal rings.\n\n## Explanations\n\n### L1 human\n\nAn element is integral over $A$ if it satisfies an equation $b^n + a_{n-1} b^{n-1} + \\cdots + a_0 = 0$ with all $a_i$ in $A$.\nThe number $\\sqrt{2}$ is integral over $\\Z$, since it is a root of $x^2 - 2$.\nThe golden ratio $(1 + \\sqrt{5})/2$ is also integral over $\\Z$, since it is a root of $x^2 - x - 1$.\nThe number $1/2$ is not integral over $\\Z$, although it is a root of $2x - 1$: the leading coefficient cannot be made $1$.\nIntegral elements behave like \"integers\" inside a larger ring.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13B21"
    ],
    "name": "Integral element",
    "same_as": {
      "wikidata": "Q1493740",
      "wikipedia": "Integral element"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Integral_element"
      }
    ],
    "tag": "1EHN4KPSFS"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/1EHN4KPSFS/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/1EHN4KPSFS.md",
  "msc": [
    "13B21"
  ],
  "name": "Integral element",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1EH/1EHN4KPSFS.md",
  "status": null,
  "tag": "1EHN4KPSFS",
  "uri": "m2://1EHN4KPSFS",
  "url": "https://math2.tamnd.com/wiki/integral-element/",
  "writing": {
    "tag": "1EHN4KPSFS",
    "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
      }
    ]
  }
}