{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $K$ and $L$ be [[fields|0QA8CHH8MG]].\nA field homomorphism is a map $f \\colon K \\to L$ such that:\n\n1. $f(a + b) = f(a) + f(b)$ for all $a, b \\in K$;\n2. $f(ab) = f(a) f(b)$ for all $a, b \\in K$;\n3. $f(1) = 1$.\n\n## Remarks\n\nSo a field homomorphism is a [[ring homomorphism|0GGE0Z0JQ1]] between two fields.\nExample: complex conjugation $f(a + bi) = a - bi$ on $\\C$ satisfies the three clauses, since conjugation respects sums and products and fixes $1$.\nNon-example: $f(a) = 2a$ on $\\Q$ is additive, but $f(1) = 2 \\ne 1$ and $f(1 \\cdot 1) = 2 \\ne 4 = f(1) f(1)$.\nEvery field homomorphism is injective, by [[ring homomorphisms from a field are injective|09N7DZ7SDB]].\nSo $f(K)$ is a [[subfield|07QY5RPFBR]] of $L$ isomorphic to $K$, and $f$ is also called an embedding of $K$ into $L$.\nSince $f(n \\cdot 1) = n \\cdot 1$ and $f$ is injective, $K$ and $L$ have the same [[characteristic|1ZB13N78XW]].\nA bijective field homomorphism is a field isomorphism.\n\n## Explanations\n\n### L1 human\n\nA field homomorphism is a map that keeps sums, products and the number $1$.\nBecause a field has inverses, such a map cannot collapse anything.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "EZVN87C88J",
    "kind": "definition",
    "msc": [
      "12F05",
      "12E99"
    ],
    "name": "Field homomorphism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Field_(mathematics)"
      }
    ],
    "tag": "JBZMRZY4YF",
    "uses": [
      "0QA8CHH8MG",
      "0GGE0Z0JQ1"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/JBZMRZY4YF/",
  "links": [
    {
      "field": "defines",
      "target": "EZVN87C88J"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "0GGE0Z0JQ1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JBZMRZY4YF.md",
  "msc": [
    "12F05",
    "12E99"
  ],
  "name": "Field homomorphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JBZ/JBZMRZY4YF.md",
  "status": null,
  "tag": "JBZMRZY4YF",
  "uri": "m2://JBZMRZY4YF",
  "url": "https://math2.tamnd.com/wiki/field-homomorphism-definition/",
  "writing": {
    "tag": "JBZMRZY4YF",
    "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
      }
    ]
  }
}