{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\overline{\\mathbb{Q}}$ be the set of the complex numbers that are [[algebraic|0NPW1DWECM]] over $\\mathbb{Q}$.\nThen $\\overline{\\mathbb{Q}}$ is a [[subfield|07QY5RPFBR]] of $\\mathbb{C}$.\n\n## Remarks\n\nThe proof: if $a, b$ are algebraic over $\\mathbb{Q}$, then $\\mathbb{Q}(a, b)$ is a finite extension of $\\mathbb{Q}$, by the [[tower law|0A7KJ68XHT]]. So $a + b$, $a - b$, $a b$ and $a/b$ for $b \\neq 0$ are algebraic, because [[finite extensions are algebraic|0DFYTNZ6AM]].\nMore generally, for every extension $L/K$, the elements of $L$ that are algebraic over $K$ form a subfield of $L$.\nThe field $\\overline{\\mathbb{Q}}$ is countable and algebraically closed. It is an [[algebraic closure|0MCXX29HWY]] of $\\mathbb{Q}$.\n\n## Explanations\n\n### L1 human\n\nThe sum and the product of two algebraic numbers are algebraic.\nFor example, $\\sqrt{2} + \\sqrt{3}$ is a root of $x^4 - 10x^2 + 1$.\nThe proof does not need to find such a polynomial: it uses only dimensions of vector spaces.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "11R04"
    ],
    "name": "The algebraic numbers form a field",
    "role": "theorem",
    "sources": [
      {
        "locator": "Field",
        "url": "https://en.wikipedia.org/wiki/Algebraic_number"
      },
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Algebraic_element"
      }
    ],
    "tag": "0QFTETCHKB",
    "uses": [
      "0NPW1DWECM",
      "0A7KJ68XHT",
      "0DFYTNZ6AM",
      "07QY5RPFBR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0QFTETCHKB/",
  "links": [
    {
      "field": "uses",
      "target": "0NPW1DWECM"
    },
    {
      "field": "uses",
      "target": "0A7KJ68XHT"
    },
    {
      "field": "uses",
      "target": "0DFYTNZ6AM"
    },
    {
      "field": "uses",
      "target": "07QY5RPFBR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0QFTETCHKB.md",
  "msc": [
    "11R04"
  ],
  "name": "The algebraic numbers form a field",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0QF/0QFTETCHKB.md",
  "status": null,
  "tag": "0QFTETCHKB",
  "uri": "m2://0QFTETCHKB",
  "url": "https://math2.tamnd.com/wiki/the-algebraic-numbers-form-a-field/",
  "writing": {
    "tag": "0QFTETCHKB",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}