{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "SVHTAFD07A"
    }
  ],
  "body": "\n## Summary\n\nKummer–Artin–Schreier–Witt theory describes the [[abelian extensions|EQDEXBWGJH]] of bounded exponent of a [[field|04C5ZBG1WT]] by explicit equations.\nFor an exponent $m$ prime to the [[characteristic|1ZB13N78XW]], this is [[Kummer theory|XZQENYV15Y]].\nIt uses $m$-th roots and needs the $m$-th [[roots of unity|0YV2CKP2AV]] in the base field.\nFor an exponent $p^n$ in characteristic $p \u003e 0$, this is Artin–Schreier–Witt theory.\nIt uses equations $F(x) - x = a$ in the ring of [[Witt vectors|6N3X2QX4W9]] of length $n$.\nFor $n = 1$ it reduces to [[Artin–Schreier theory|7YNMNRGH3T]].\nTogether the two parts describe all finite cyclic extensions of a field that contains enough roots of unity.\n\n## Explanations\n\n### L1 human\n\nKummer theory builds extensions by adjoining roots such as $\\sqrt[m]{a}$.\nIn characteristic $p$ this fails for $m = p$, because $x^p - a$ has only one root, repeated.\nArtin and Schreier replaced $x^p = a$ by $x^p - x = a$, which gives the cyclic extensions of degree $p$.\nWitt found that to reach cyclic extensions of degree $p^2, p^3, \\ldots$ one must solve the same kind of equation with Witt vectors in place of field elements.\nFor example, over the finite field $\\F_p$ the Frobenius map fixes every Witt vector, because $c^p = c$ for each $c \\in \\F_p$.\nFrom this the theory recovers that $\\F_p$ has exactly one extension of each degree $p^n$, and that it is cyclic.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "12F10",
      "13F35",
      "12G05"
    ],
    "name": "Kummer–Artin–Schreier–Witt theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Witt_vector"
      }
    ],
    "tag": "R45FHBE15X"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/R45FHBE15X/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/R45FHBE15X.md",
  "msc": [
    "12F10",
    "13F35",
    "12G05"
  ],
  "name": "Kummer–Artin–Schreier–Witt theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R45/R45FHBE15X.md",
  "status": null,
  "tag": "R45FHBE15X",
  "uri": "m2://R45FHBE15X",
  "url": "https://tamnd.github.io/math2/wiki/kummer-artin-schreier-witt-theory/",
  "writing": {
    "tag": "R45FHBE15X",
    "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
      }
    ]
  }
}