{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nIt is a *radical extension* if there are elements $\\alpha_1, \\dots, \\alpha_m \\in L$ and positive integers $n_1, \\dots, n_m$ such that:\n\n1. $L = K(\\alpha_1, \\dots, \\alpha_m)$;\n2. $\\alpha_1^{n_1} \\in K$;\n3. $\\alpha_i^{n_i} \\in K(\\alpha_1, \\dots, \\alpha_{i-1})$ for each $i$ with $2 \\le i \\le m$.\n\nEquivalently, there is a tower $K = K_0 \\subseteq K_1 \\subseteq \\dots \\subseteq K_m = L$ in which each $K_i = K_{i-1}(\\alpha_i)$ is a [[simple extension|0P6SBR7YM2]] with $\\alpha_i^{n_i} \\in K_{i-1}$.\n\n## Remarks\n\nExample: $L = \\Q(\\sqrt{2}, \\sqrt[3]{1 + \\sqrt{2}})$ is radical over $\\Q$, with $\\alpha_1 = \\sqrt{2}$, $n_1 = 2$, $\\alpha_2 = \\sqrt[3]{1 + \\sqrt{2}}$, $n_2 = 3$, and $\\alpha_2^3 = 1 + \\sqrt{2} \\in \\Q(\\alpha_1)$.\nNon-example: let $\\theta = \\cos(2\\pi/7)$, a root of the irreducible cubic $8x^3 + 4x^2 - 4x - 1$.\nThe field $\\Q(\\theta)$ is real, and its three conjugates $\\cos(2\\pi k/7)$, $k = 1, 2, 3$, are real, so by the [[casus irreducibilis|MJ864PBRRJ]] $\\theta$ lies in no radical extension of $\\Q$ contained in $\\R$.\nIn particular $\\Q(\\theta)/\\Q$ is not radical, although $\\Q(\\theta)$ lies inside the radical extension $\\Q(\\zeta_7)$, with $\\zeta_7 = e^{2\\pi i/7}$.\nA radical extension is finite, but it need not be normal or separable: $\\Q(\\sqrt[3]{2})/\\Q$ is radical and not normal.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "V74WMGQQ9D",
    "kind": "definition",
    "msc": [
      "12F10",
      "12F05"
    ],
    "name": "Radical extension",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Radical_extension"
      }
    ],
    "tag": "7FYMK2XH1V",
    "uses": [
      "0WQ1A8SHTZ",
      "0P6SBR7YM2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/7FYMK2XH1V/",
  "links": [
    {
      "field": "defines",
      "target": "V74WMGQQ9D"
    },
    {
      "field": "uses",
      "target": "0WQ1A8SHTZ"
    },
    {
      "field": "uses",
      "target": "0P6SBR7YM2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7FYMK2XH1V.md",
  "msc": [
    "12F10",
    "12F05"
  ],
  "name": "Radical extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7FY/7FYMK2XH1V.md",
  "status": null,
  "tag": "7FYMK2XH1V",
  "uri": "m2://7FYMK2XH1V",
  "url": "https://math2.tamnd.com/wiki/radical-extension-definition/",
  "writing": {
    "tag": "7FYMK2XH1V",
    "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
      }
    ]
  }
}