{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0CBRFQ7DVZ"
    },
    {
      "field": "uses",
      "target": "0DFYTNZ6AM"
    },
    {
      "field": "uses",
      "target": "0MCXX29HWY"
    },
    {
      "field": "uses",
      "target": "0N019F6RE9"
    },
    {
      "field": "uses",
      "target": "0TPWS0K1BC"
    }
  ],
  "body": "\n## Definition\n\nA [[field extension|0WQ1A8SHTZ]] $L/K$ is algebraic if every element of $L$ is [[algebraic|0NPW1DWECM]] over $K$.\n\n## Remarks\n\n[[Every finite extension is algebraic|0DFYTNZ6AM]], so $\\mathbb{C}/\\mathbb{R}$ and $\\mathbb{Q}(\\sqrt{2})/\\mathbb{Q}$ are algebraic.\nThe converse fails: the field of all algebraic numbers is an algebraic extension of $\\mathbb{Q}$ of infinite degree.\nThe extension $\\mathbb{R}/\\mathbb{Q}$ is not algebraic, because $\\pi$ is transcendental.\nAn [[algebraically closed field|02SJG1HC32]] has no algebraic extension other than itself.\n\n## Explanations\n\n### L1 human\n\nIn an algebraic extension, every new element solves a polynomial equation over the base field.\nThere are no elements that behave like free variables.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0PTBM7K0R2",
    "kind": "definition",
    "msc": [
      "12F05"
    ],
    "name": "Algebraic extension",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Algebraic_extension"
      }
    ],
    "tag": "09EYMKMSB9",
    "uses": [
      "0WQ1A8SHTZ",
      "0NPW1DWECM"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/09EYMKMSB9/",
  "links": [
    {
      "field": "defines",
      "target": "0PTBM7K0R2"
    },
    {
      "field": "uses",
      "target": "0WQ1A8SHTZ"
    },
    {
      "field": "uses",
      "target": "0NPW1DWECM"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09EYMKMSB9.md",
  "msc": [
    "12F05"
  ],
  "name": "Algebraic extension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09E/09EYMKMSB9.md",
  "status": null,
  "tag": "09EYMKMSB9",
  "uri": "m2://09EYMKMSB9",
  "url": "https://math2.tamnd.com/wiki/algebraic-extension-definition/",
  "writing": {
    "tag": "09EYMKMSB9",
    "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
      }
    ]
  }
}