{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0SSSM1D475"
    }
  ],
  "body": "\n## Statement\n\nThe set of the [[constructible numbers|0FMN2MN7TM]] is a [[subfield|07QY5RPFBR]] of $\\mathbb{C}$.\nIf $z$ is constructible, then the square roots of $z$ are constructible.\nMoreover, the field of constructible numbers is the smallest subfield of $\\mathbb{C}$ that is closed under square roots.\n\n## Remarks\n\nThe proof of closure uses classical constructions: parallels and similar triangles give products and quotients, and a circle on a diameter of length $1 + x$ gives $\\sqrt{x}$.\nConversely, the intersection points of lines and circles have coordinates that come from the old ones by field operations and square roots.\nSo a number is constructible if and only if it is obtained from $\\mathbb{Q}$ by finitely many field operations and square roots.\nThe constructible real numbers form an ordered field in which every positive element has a square root.\n\n## Explanations\n\n### L1 human\n\nYou can add, subtract, multiply and divide lengths with straightedge and compass, and you can take square roots.\nNothing else is possible, because lines and circles only lead to equations of degree one or two.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "12F05"
    ],
    "name": "The constructible numbers form a field closed under square roots",
    "role": "theorem",
    "sources": [
      {
        "locator": "Algebraic properties",
        "url": "https://en.wikipedia.org/wiki/Constructible_number"
      },
      {
        "locator": "Equivalence of algebraic and geometric definitions",
        "url": "https://en.wikipedia.org/wiki/Constructible_number"
      }
    ],
    "tag": "0PN9WMRWT7",
    "uses": [
      "0FMN2MN7TM",
      "07QY5RPFBR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0PN9WMRWT7/",
  "links": [
    {
      "field": "uses",
      "target": "0FMN2MN7TM"
    },
    {
      "field": "uses",
      "target": "07QY5RPFBR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0PN9WMRWT7.md",
  "msc": [
    "12F05"
  ],
  "name": "The constructible numbers form a field closed under square roots",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0PN/0PN9WMRWT7.md",
  "status": null,
  "tag": "0PN9WMRWT7",
  "uri": "m2://0PN9WMRWT7",
  "url": "https://math2.tamnd.com/wiki/the-constructible-numbers-form-a-field-closed-under-square-roots/",
  "writing": {
    "tag": "0PN9WMRWT7",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}