{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "09RFKMBQ63"
    }
  ],
  "body": "\n## Statement\n\nThe [[rational numbers|0NCJ5WXN6Z]] $\\mathbb{Q}$, with their addition and multiplication, form a field.\nThat is, $\\mathbb{Q}$ is a commutative ring with $0 \\neq 1$, and each rational number $q \\neq 0$ has a multiplicative inverse $q^{-1}$ with $q \\cdot q^{-1} = 1$.\n\n## Remarks\n\nThe inverse of $a / b$ with $a \\neq 0$ is $b / a$.\nThe field $\\mathbb{Q}$ is the smallest subfield of the real numbers. Each field of characteristic $0$ contains a copy of $\\mathbb{Q}$.\nThe [[integers|0C4JS9RF5M]] are not a field, and $\\mathbb{Q}$ is the field of fractions of $\\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nIn the rational numbers, you can add, subtract, multiply, and divide by any number that is not zero.\nThe result is always a rational number again.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "12E99"
    ],
    "name": "The rationals form a field",
    "role": "theorem",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Rational_number"
      }
    ],
    "tag": "0N2WPCC00T",
    "uses": [
      "0NCJ5WXN6Z",
      "0BXA4Q0JHW"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0N2WPCC00T/",
  "links": [
    {
      "field": "uses",
      "target": "0NCJ5WXN6Z"
    },
    {
      "field": "uses",
      "target": "0BXA4Q0JHW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0N2WPCC00T.md",
  "msc": [
    "12E99"
  ],
  "name": "The rationals form a field",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0N2/0N2WPCC00T.md",
  "status": null,
  "tag": "0N2WPCC00T",
  "uri": "m2://0N2WPCC00T",
  "url": "https://math2.tamnd.com/wiki/the-rationals-form-a-field/",
  "writing": {
    "tag": "0N2WPCC00T",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}