{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L/K$ be an [[algebraic extension|09EYMKMSB9]].\nLet $\\Omega$ be an [[algebraically closed field|02SJG1HC32]] and let $\\sigma \\colon K \\to \\Omega$ be a [[field homomorphism|EZVN87C88J]].\nThe *separable degree* $[L:K]_s$ of $L/K$ is the cardinality of the set of field homomorphisms $\\tau \\colon L \\to \\Omega$ with $\\tau|_K = \\sigma$.\nThis cardinality does not depend on the choice of $\\Omega$ and $\\sigma$.\n\n## Remarks\n\nLet $L/K$ be finite.\nThen $[L:K]_s$ divides the [[degree|0GS74FGFEK]] $[L:K]$.\nIf $K$ has characteristic $p \u003e 0$, the quotient $[L:K]_i = [L:K] / [L:K]_s$ is a power of $p$, called the inseparable degree; in characteristic $0$ it is $1$.\nThe extension $L/K$ is separable if and only if $[L:K]_s = [L:K]$.\nSeparable degree is multiplicative in towers: $[M:K]_s = [M:L]_s [L:K]_s$.\nIt equals the degree over $K$ of the [[separable closure|JJ91A1V8NQ]] of $K$ in $L$.\n\nExample: $L = \\Q(\\sqrt{2})$ over $\\Q$ with $\\Omega = \\C$.\nCheck: an embedding is determined by the image of $\\sqrt{2}$, which must be a root of $x^2 - 2$, so there are exactly $2$ embeddings and $[L:\\Q]_s = 2 = [L:\\Q]$.\n\nNon-example of the equality $[L:K]_s = [L:K]$: let $K = \\F_p(t)$ and $L = K(s)$ with $s^p = t$.\nOver an algebraic closure, $x^p - t = (x - s)^p$ has one root, so $[L:K]_s = 1$ while $[L:K] = p$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "H5YX5B7GVM",
    "kind": "definition",
    "msc": [
      "12F10",
      "12F15"
    ],
    "name": "Separable degree",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Separable_extension"
      }
    ],
    "tag": "XS3VFNSY9N",
    "uses": [
      "09EYMKMSB9",
      "02SJG1HC32",
      "EZVN87C88J",
      "0GS74FGFEK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/XS3VFNSY9N/",
  "links": [
    {
      "field": "defines",
      "target": "H5YX5B7GVM"
    },
    {
      "field": "uses",
      "target": "09EYMKMSB9"
    },
    {
      "field": "uses",
      "target": "02SJG1HC32"
    },
    {
      "field": "uses",
      "target": "EZVN87C88J"
    },
    {
      "field": "uses",
      "target": "0GS74FGFEK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/XS3VFNSY9N.md",
  "msc": [
    "12F10",
    "12F15"
  ],
  "name": "Separable degree",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XS3/XS3VFNSY9N.md",
  "status": null,
  "tag": "XS3VFNSY9N",
  "uri": "m2://XS3VFNSY9N",
  "url": "https://math2.tamnd.com/wiki/separable-degree-definition/",
  "writing": {
    "tag": "XS3VFNSY9N",
    "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
      }
    ]
  }
}