{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "05K3J8J8T2"
    },
    {
      "field": "uses",
      "target": "0QFV5Y1Z5X"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]].\nOn the set of pairs $(a, b)$ with $a, b \\in R$ and $b \\neq 0$, let $(a, b) \\sim (c, d)$ if $a d = b c$.\nThis is an equivalence relation. Write $a/b$ for the class of $(a, b)$.\nThe field of fractions $\\operatorname{Frac}(R)$ is the set of these classes, with the operations\n$$\\frac{a}{b} + \\frac{c}{d} = \\frac{a d + b c}{b d}, \\qquad \\frac{a}{b} \\cdot \\frac{c}{d} = \\frac{a c}{b d}.$$\nIt is a [[field|0QA8CHH8MG]], and the map $R \\to \\operatorname{Frac}(R)$, $a \\mapsto a/1$, is an injective ring homomorphism.\n\n## Remarks\n\nThe product $b d$ is not zero because $R$ is an integral domain. This makes the operations well defined.\nThe inverse of a nonzero $a/b$ is $b/a$.\nUniversal property: every injective ring homomorphism from $R$ to a field $K$ extends to exactly one ring homomorphism from $\\operatorname{Frac}(R)$ to $K$.\nThe field of fractions of the polynomial ring $K[x]$ over a field $K$ is the field $K(x)$ of rational functions.\n\n## Explanations\n\n### L1 human\n\nThis is how the rational numbers come from the integers.\nA fraction is a pair of a numerator and a nonzero denominator.\nTwo fractions are equal when cross-multiplication gives the same result, as with $1/2 = 2/4$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0F2PTGX1A2",
    "kind": "definition",
    "msc": [
      "13B30"
    ],
    "name": "Field of fractions",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Field_of_fractions"
      }
    ],
    "tag": "0VF11ZTTMD",
    "uses": [
      "09S3PFBEPM",
      "0QA8CHH8MG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0VF11ZTTMD/",
  "links": [
    {
      "field": "defines",
      "target": "0F2PTGX1A2"
    },
    {
      "field": "uses",
      "target": "09S3PFBEPM"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0VF11ZTTMD.md",
  "msc": [
    "13B30"
  ],
  "name": "Field of fractions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0VF/0VF11ZTTMD.md",
  "status": null,
  "tag": "0VF11ZTTMD",
  "uri": "m2://0VF11ZTTMD",
  "url": "https://math2.tamnd.com/wiki/field-of-fractions-definition/",
  "writing": {
    "tag": "0VF11ZTTMD",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}