{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "72W4HXM98H"
    },
    {
      "field": "uses",
      "target": "AA943HVK6Z"
    },
    {
      "field": "uses",
      "target": "E7WHZXQ0WF"
    },
    {
      "field": "uses",
      "target": "N36F259F1B"
    },
    {
      "field": "uses",
      "target": "REJDQ6BTQV"
    },
    {
      "field": "uses",
      "target": "X8VKW5Y1CZ"
    },
    {
      "field": "about",
      "target": "REJDQ6BTQV"
    }
  ],
  "body": "\n## Summary\n\nLet $L/K$ be an extension of [[algebraic number fields|MWBDFS2KJ5]].\nA [[prime ideal|00MHHZ9SNE]] of the ring of integers of $K$ need not stay prime in the ring of integers of $L$.\nIt factors into a product of prime ideals of $L$.\nThe factorisation is described by the number of factors, their exponents, called [[ramification|H5RC67SD24]] indices, and their residue degrees.\nThese numbers satisfy the [[fundamental identity|REJDQ6BTQV]] $\\sum_i e_i f_i = [L : K]$.\nIn a Galois extension the splitting is controlled by the [[decomposition group|7X92G8CQ33]] and the [[Frobenius element|H5145REFMW]].\nThis leads to [[Chebotarev's density theorem|X8VKW5Y1CZ]] and to [[class field theory|BEKCEYHCHX]].\n\n## Explanations\n\n### L1 human\n\nA prime number can stop being prime in a bigger ring of integers.\nIn the Gaussian integers $\\Z[i]$, the prime $5$ splits as $(2 + i)(2 - i)$, the prime $3$ stays prime, and $2 = -i(1 + i)^2$ becomes a square up to a unit.\nThese three behaviours are called splitting, staying inert and ramifying.\nWhich one happens depends on $p$ modulo $4$: odd primes $p \\equiv 1 \\pmod 4$ split and primes $p \\equiv 3 \\pmod 4$ stay inert.\nSuch rules for general extensions are a central topic of algebraic number theory.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "11R04",
      "11R32"
    ],
    "name": "Splitting of prime ideals",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions"
      }
    ],
    "tag": "B7KNZCBFYT"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/B7KNZCBFYT/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/B7KNZCBFYT.md",
  "msc": [
    "11R04",
    "11R32"
  ],
  "name": "Splitting of prime ideals",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/B7K/B7KNZCBFYT.md",
  "status": null,
  "tag": "B7KNZCBFYT",
  "uri": "m2://B7KNZCBFYT",
  "url": "https://math2.tamnd.com/wiki/splitting-of-prime-ideals/",
  "writing": {
    "tag": "B7KNZCBFYT",
    "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
      }
    ]
  }
}