{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0VP4TBVJ19"
    }
  ],
  "body": "\n## Definition\n\nLet $p$ be a prime number and let $R$ be a [[commutative ring|0GYGHKX834]] of [[characteristic|0P7CH1CND5]] $p$.\nThe Frobenius endomorphism of $R$ is the map $F \\colon R \\to R$, $F(x) = x^p$.\nIt is a [[ring homomorphism|0GGE0Z0JQ1]] from $R$ to $R$.\n\n## Remarks\n\nThe map $F$ preserves products because $R$ is commutative. It preserves sums by the [[freshman's dream|0396VGFA78]].\nIf $R$ is an [[integral domain|09S3PFBEPM]], then $F$ is injective.\nIf $R$ is a [[finite field|08RTDHX1SK]], then $F$ is an automorphism of $R$.\nOn $\\mathbb{F}_p$, the map $F$ is the identity, by Fermat's little theorem.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$, raising to the $p$-th power respects addition and multiplication.\nSo it is a homomorphism from the ring to itself.\nThis simple map is a basic tool in the study of finite fields.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0T3WVTD675",
    "kind": "definition",
    "msc": [
      "13A35"
    ],
    "name": "Frobenius endomorphism",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Frobenius_endomorphism"
      }
    ],
    "tag": "0GHDDVAH8P",
    "uses": [
      "0GYGHKX834",
      "0P7CH1CND5",
      "0396VGFA78",
      "0GGE0Z0JQ1"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0GHDDVAH8P/",
  "links": [
    {
      "field": "defines",
      "target": "0T3WVTD675"
    },
    {
      "field": "uses",
      "target": "0GYGHKX834"
    },
    {
      "field": "uses",
      "target": "0P7CH1CND5"
    },
    {
      "field": "uses",
      "target": "0396VGFA78"
    },
    {
      "field": "uses",
      "target": "0GGE0Z0JQ1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0GHDDVAH8P.md",
  "msc": [
    "13A35"
  ],
  "name": "Frobenius endomorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0GH/0GHDDVAH8P.md",
  "status": null,
  "tag": "0GHDDVAH8P",
  "uri": "m2://0GHDDVAH8P",
  "url": "https://math2.tamnd.com/wiki/frobenius-endomorphism-definition/",
  "writing": {
    "tag": "0GHDDVAH8P",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}