{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $K^s$ be a [[separable closure|JJ91A1V8NQ]] of $K$.\n*Kummer–Artin–Schreier–Witt theory* is the description of the finite [[abelian extensions|EQDEXBWGJH]] of $K$ inside $K^s$ of exponent dividing a given integer.\nIt has two parts.\n\nThe Kummer part is [[Kummer theory|XZQENYV15Y]].\nIt applies to an integer $m \\ge 1$ not divisible by the [[characteristic|1ZB13N78XW]] of $K$, when $K$ contains $m$ distinct $m$-th roots of unity.\nIt describes the abelian extensions of exponent dividing $m$ by adjoining $m$-th roots.\n\nThe Artin–Schreier–Witt part concerns a field $K$ of characteristic $p \u003e 0$ and an integer $n \\ge 1$.\nLet $W_n(K)$ be the ring of [[Witt vectors|6N3X2QX4W9]] of length $n$.\nIts elements are sequences $(a_0, \\ldots, a_{n-1})$ in $K$, with the first $n$ Witt sum and product polynomials as operations.\nLet $F(a_0, \\ldots, a_{n-1}) = (a_0^p, \\ldots, a_{n-1}^p)$ be the [[Frobenius|0T3WVTD675]] endomorphism of $W_n(K)$, and let $\\wp = F - \\id$, an additive homomorphism.\nFor a group $\\Delta$ with $\\wp W_n(K) \\subseteq \\Delta \\subseteq W_n(K)$, let $K_\\Delta$ be the field generated over $K$ by the entries of all $x \\in W_n(K^s)$ with $\\wp(x) \\in \\Delta$.\nThe theory states:\n\n1. the map $\\wp \\colon W_n(K^s) \\to W_n(K^s)$ is surjective, and its kernel is $W_n(\\F_p) \\cong \\Z/p^n\\Z$;\n2. the map $\\Delta \\mapsto K_\\Delta$ is a bijection from the groups $\\Delta$ as above with $\\Delta / \\wp W_n(K)$ finite onto a set of extensions.\n   This set consists of the finite abelian extensions of $K$ inside $K^s$ whose [[Galois group|0XTY86BZKY]] has exponent dividing $p^n$;\n3. for such $\\Delta$, the pairing $\\operatorname{Gal}(K_\\Delta/K) \\times \\Delta/\\wp W_n(K) \\to \\Z/p^n\\Z$, $(\\sigma, a) \\mapsto \\sigma(x) - x$ with $\\wp(x) = a$, is well defined and perfect, so $\\operatorname{Gal}(K_\\Delta/K) \\cong \\Hom(\\Delta/\\wp W_n(K), \\Z/p^n\\Z)$.\n\n## Remarks\n\nFor $n = 1$, $W_1(K) = K$ and $\\wp(x) = x^p - x$, so the Artin–Schreier–Witt part is [[Artin–Schreier theory|7YNMNRGH3T]].\nThe cyclic extensions of degree $p^n$ correspond to the groups $\\Delta$ generated by $\\wp W_n(K)$ and one $a$ whose class has order $p^n$.\n\nExample: take $K = \\F_p$ and $n = 2$.\nHere $F$ is the identity on $W_2(\\F_p)$, because $c^p = c$ for $c \\in \\F_p$, so $\\wp W_2(\\F_p) = 0$.\nThus $W_2(\\F_p)/\\wp W_2(\\F_p) \\cong \\Z/p^2\\Z$ is cyclic of order $p^2$.\nBy clauses 2 and 3, $\\F_p$ has exactly one extension in $K^s$ with Galois group of exponent dividing $p^2$ and of degree $p^2$, and it is cyclic: it is $\\F_{p^{p^2}}$.\n\nNon-example: the Kummer part needs the roots of unity in the base field.\nOver $\\Q$ with $m = 3$, the field $\\Q(\\sqrt[3]{2})$ is not Galois over $\\Q$, since its other conjugates $\\omega \\sqrt[3]{2}$, with $\\omega$ a primitive cube root of unity, are not real.\nSo adjoining $m$-th roots does not give abelian extensions when $K$ lacks the $m$-th roots of unity.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "R45FHBE15X",
    "kind": "definition",
    "msc": [
      "12F10",
      "13F35",
      "12G05"
    ],
    "name": "Kummer–Artin–Schreier–Witt theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Witt_vector"
      }
    ],
    "tag": "SVHTAFD07A",
    "uses": [
      "0QA8CHH8MG",
      "1ZB13N78XW",
      "6N3X2QX4W9",
      "JJ91A1V8NQ",
      "EQDEXBWGJH",
      "0XTY86BZKY",
      "0T3WVTD675",
      "XZQENYV15Y",
      "7YNMNRGH3T"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/SVHTAFD07A/",
  "links": [
    {
      "field": "defines",
      "target": "R45FHBE15X"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "1ZB13N78XW"
    },
    {
      "field": "uses",
      "target": "6N3X2QX4W9"
    },
    {
      "field": "uses",
      "target": "JJ91A1V8NQ"
    },
    {
      "field": "uses",
      "target": "EQDEXBWGJH"
    },
    {
      "field": "uses",
      "target": "0XTY86BZKY"
    },
    {
      "field": "uses",
      "target": "0T3WVTD675"
    },
    {
      "field": "uses",
      "target": "XZQENYV15Y"
    },
    {
      "field": "uses",
      "target": "7YNMNRGH3T"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SVHTAFD07A.md",
  "msc": [
    "12F10",
    "13F35",
    "12G05"
  ],
  "name": "Kummer–Artin–Schreier–Witt theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SVH/SVHTAFD07A.md",
  "status": null,
  "tag": "SVHTAFD07A",
  "uri": "m2://SVHTAFD07A",
  "url": "https://tamnd.github.io/math2/wiki/kummer-artin-schreier-witt-theory-definition/",
  "writing": {
    "tag": "SVHTAFD07A",
    "kind": "definition",
    "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
      }
    ]
  }
}