{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "218WAD58W3"
    },
    {
      "field": "uses",
      "target": "5Q5VN56QH4"
    },
    {
      "field": "uses",
      "target": "BEEAETX7SB"
    },
    {
      "field": "uses",
      "target": "CKGN4ZH6P2"
    },
    {
      "field": "uses",
      "target": "DPFMXS71D1"
    },
    {
      "field": "uses",
      "target": "EESNTXFWHZ"
    },
    {
      "field": "defines",
      "target": "GGCHBSY4HK"
    }
  ],
  "body": "\n## Summary\n\nThe separable closure of $K$ in an algebraic extension $L$ is the set of elements of $L$ that are [[separable|0WCX7NB4N7]] over $K$.\nIt is the largest separable subextension of $L/K$.\nThe separable closure of a field $K$, written $K^{\\mathrm{sep}}$, is the separable closure of $K$ in an [[algebraic closure|0X6AB5P3G1]]; it is a [[separably closed field|8B57B7EGK4]].\nIts automorphism group over $K$ is the [[absolute Galois group|PGKWBJDTKX]] of $K$.\nFor a [[perfect field|VWBVKT794W]] the separable closure is the whole algebraic closure.\n\n## Explanations\n\n### L1 human\n\nInside a big field, collect all the elements whose minimal polynomials over $K$ have no repeated roots.\nThese elements form a field, and it is the separable closure.\nIn characteristic $0$ every polynomial behaves well, so nothing is lost and one gets the algebraic closure.\nIn characteristic $p$, elements such as a $p$-th root of $t$ over $\\F_p(t)$ are left out.\nThe separable closure is the part of the algebraic closure that Galois theory sees.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "12F10"
    ],
    "name": "Separable closure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Separable_extension"
      }
    ],
    "tag": "JJ91A1V8NQ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/JJ91A1V8NQ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/JJ91A1V8NQ.md",
  "msc": [
    "12F10"
  ],
  "name": "Separable closure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JJ9/JJ91A1V8NQ.md",
  "status": null,
  "tag": "JJ91A1V8NQ",
  "uri": "m2://JJ91A1V8NQ",
  "url": "https://math2.tamnd.com/wiki/separable-closure/",
  "writing": {
    "tag": "JJ91A1V8NQ",
    "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
      }
    ]
  }
}