{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "YJK4WF6YWV"
    }
  ],
  "body": "\n## Summary\n\nThe ramification groups of a finite Galois extension of local fields form a decreasing chain of normal subgroups $G_0 \\supseteq G_1 \\supseteq G_2 \\supseteq \\cdots$ of its [[Galois group|0XTY86BZKY]].\nThe group $G_0$ is the inertia group, and $G_1$ is the wild inertia group.\nThe chain measures how deeply the extension is [[ramified|H5RC67SD24]].\nIt controls the different and the conductor of the extension.\n\n## Explanations\n\n### L1 human\n\nAn automorphism of a field extension moves elements of the ring of integers.\nSome automorphisms move every integer only a little, in the sense of the valuation.\nThe group $G_i$ collects the automorphisms that move every integer by an amount divisible by the $(i+1)$-st power of a uniformizer.\nThe larger $i$ is, the closer to the identity these automorphisms are.\nFor the extension $\\Q_2(i) / \\Q_2$, complex conjugation lies in $G_0$ and $G_1$, but not in $G_2$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11S15",
      "11S20"
    ],
    "name": "Ramification group",
    "same_as": {
      "wikidata": "Q7289549",
      "wikipedia": "Ramification group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ramification_group"
      }
    ],
    "specializes": [
      "0X1JXT47GA"
    ],
    "tag": "BQJ9ZX9P3V"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/BQJ9ZX9P3V/",
  "links": [
    {
      "field": "specializes",
      "target": "0X1JXT47GA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/BQJ9ZX9P3V.md",
  "msc": [
    "11S15",
    "11S20"
  ],
  "name": "Ramification group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BQJ/BQJ9ZX9P3V.md",
  "status": null,
  "tag": "BQJ9ZX9P3V",
  "uri": "m2://BQJ9ZX9P3V",
  "url": "https://math2.tamnd.com/wiki/ramification-group/",
  "writing": {
    "tag": "BQJ9ZX9P3V",
    "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
      }
    ]
  }
}