{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]], let $a \\in K$ and let $n \\ge 2$ be an integer.\nThe polynomial $x^n - a$ is [[irreducible|00X2A9G9W3]] in $K[x]$ if and only if both of the following hold:\n\n1. for every prime $p$ dividing $n$, $a$ is not of the form $b^p$ with $b \\in K$;\n2. if $4$ divides $n$, then $a$ is not of the form $-4 c^4$ with $c \\in K$.\n\n## Remarks\n\nThe conditions are necessary: if $a = b^p$ with $p \\mid n$, then $y^p - b^p$ has the factor $y - b$, and $y = x^{n/p}$ gives a factor of $x^n - a$.\nIf $a = -4c^4$ and $4 \\mid n$, the identity $y^4 + 4c^4 = (y^2 + 2cy + 2c^2)(y^2 - 2cy + 2c^2)$ with $y = x^{n/4}$ gives a factorisation.\nThe proof of sufficiency first treats $n$ prime with a norm argument, then prime powers, and combines them with the multiplicativity of degrees in towers of extensions.\nExample: $x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x + 2)$ over $\\Q$, although $-4$ is not a square in $\\Q$; this is the case $c = 1$ of clause 2.\nExample: $x^n - 2$ is irreducible over $\\Q$ for every $n \\ge 1$.\nFor $n$ odd, clause 2 is empty and the test reduces to $p$-th powers for the primes $p \\mid n$.\n\n## Explanations\n\n### L1 human\n\nThe polynomial $x^n - a$ factors in two obvious ways: when $a$ is a perfect $p$-th power for a prime $p$ dividing $n$, and in one special case built from $x^4 + 4$.\nCapelli's theorem says that there are no other ways.\n\n### L2 human\n\nThe theorem is the basic tool to study binomial equations $x^n = a$ and the radical extensions $K(\\sqrt[n]{a})$.\nIt gives the degree $[K(\\sqrt[n]{a}) : K] = n$ when the two clauses hold.\nThis is used in the theory of [[radical extensions|V74WMGQQ9D]] and of Kummer extensions.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "00X2A9G9W3"
    ],
    "kind": "statement",
    "msc": [
      "12E05",
      "12F05"
    ],
    "name": "Capelli's theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Irreducible_polynomial"
      }
    ],
    "tag": "HC5NDQ3DTS",
    "uses": [
      "0QA8CHH8MG",
      "00X2A9G9W3"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/HC5NDQ3DTS/",
  "links": [
    {
      "field": "about",
      "target": "00X2A9G9W3"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "00X2A9G9W3"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HC5NDQ3DTS.md",
  "msc": [
    "12E05",
    "12F05"
  ],
  "name": "Capelli's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HC5/HC5NDQ3DTS.md",
  "status": null,
  "tag": "HC5NDQ3DTS",
  "uri": "m2://HC5NDQ3DTS",
  "url": "https://math2.tamnd.com/wiki/capellis-theorem/",
  "writing": {
    "tag": "HC5NDQ3DTS",
    "kind": "statement",
    "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
      }
    ]
  }
}