{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "1TQ2F9CFVE"
    },
    {
      "field": "about",
      "target": "F99RM1T02C"
    },
    {
      "field": "uses",
      "target": "1TQ2F9CFVE"
    },
    {
      "field": "uses",
      "target": "67A4D4FAYJ"
    },
    {
      "field": "uses",
      "target": "9BDJZGZ5Z5"
    },
    {
      "field": "uses",
      "target": "F99RM1T02C"
    },
    {
      "field": "uses",
      "target": "HGK7R92QH7"
    },
    {
      "field": "uses",
      "target": "KZTFH832H0"
    },
    {
      "field": "uses",
      "target": "MRRQEEP3T8"
    },
    {
      "field": "defines",
      "target": "C7Q2MVD933"
    }
  ],
  "body": "\n## Summary\n\nThe ideal class group of a [[Dedekind domain|7NPY2CN5KB]] is the group of nonzero fractional ideals modulo the principal ones.\nThe main case is the ring of integers of a [[number field|MWBDFS2KJ5]].\nIt is trivial exactly when the ring is a [[principal ideal domain|0CMD0BCH8T]].\nFor a Dedekind domain this is the same as being a [[unique factorisation domain|0W4X9G6263]].\nSo it measures how far unique factorisation of elements fails.\nFor a number field it is a finite abelian group, and its order is the [[class number|KTKK9B1X41]].\n\n## Explanations\n\n### L1 human\n\nIn a number ring, numbers may factor in several ways, but ideals factor uniquely.\nSome ideals are generated by a single number, and those behave just like numbers; others are not.\nThe class group lumps together ideals that differ by a principal ideal and records how they multiply.\nFor $\\Z[\\sqrt{-5}]$ there are two classes: the principal ideals, and the class of the ideal $(2, 1 + \\sqrt{-5})$, whose square is the principal ideal $(2)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11R29",
      "13F05"
    ],
    "name": "Ideal class group",
    "same_as": {
      "wikidata": "Q912083",
      "wikipedia": "Ideal class group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ideal_class_group"
      }
    ],
    "tag": "WQ46BD3ARY"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/WQ46BD3ARY/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/WQ46BD3ARY.md",
  "msc": [
    "11R29",
    "13F05"
  ],
  "name": "Ideal class group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WQ4/WQ46BD3ARY.md",
  "status": null,
  "tag": "WQ46BD3ARY",
  "uri": "m2://WQ46BD3ARY",
  "url": "https://math2.tamnd.com/wiki/ideal-class-group/",
  "writing": {
    "tag": "WQ46BD3ARY",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}