{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\alpha$ and $\\beta$ be [[Gaussian integers|0B268GVX95]] with $\\beta \\neq 0$.\nThen there are Gaussian integers $q$ and $r$ with $\\alpha = q \\beta + r$ and $N(r) \u003c N(\\beta)$, where $N$ is the [[norm|00HPT88ZX0]].\nSo $\\mathbb{Z}[i]$, with the norm as Euclidean function, is a Euclidean domain.\n\n## Remarks\n\nTo find $q$, round the real part and the imaginary part of the complex number $\\alpha / \\beta$ to nearest integers. This gives even $N(r) \\le \\frac{1}{2} N(\\beta)$.\nAs a consequence, $\\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.\nThe quotient and the remainder are not always unique.\nThis is the analogue of the [[division algorithm|0CHQKZFK5S]] for integers.\n\n## History\n\nCarl Friedrich Gauss introduced the Gaussian integers in 1832, in his second paper on biquadratic reciprocity. In it, he proved unique factorization.\n\n## Explanations\n\n### L1 human\n\nYou can divide one Gaussian integer by another and get a remainder that is smaller in norm than the divisor.\nSo the Euclidean algorithm works in $\\mathbb{Z}[i]$, and factorization into primes is unique, as for the integers.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "11R11"
    ],
    "name": "The Gaussian integers form a Euclidean domain",
    "role": "theorem",
    "sources": [
      {
        "locator": "Euclidean division",
        "url": "https://en.wikipedia.org/wiki/Gaussian_integer"
      },
      {
        "locator": "Historical background",
        "url": "https://en.wikipedia.org/wiki/Gaussian_integer"
      }
    ],
    "tag": "0SPYEFH7H4",
    "uses": [
      "0B268GVX95",
      "00HPT88ZX0",
      "0CHQKZFK5S"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0SPYEFH7H4/",
  "links": [
    {
      "field": "uses",
      "target": "0B268GVX95"
    },
    {
      "field": "uses",
      "target": "00HPT88ZX0"
    },
    {
      "field": "uses",
      "target": "0CHQKZFK5S"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0SPYEFH7H4.md",
  "msc": [
    "11R11"
  ],
  "name": "The Gaussian integers form a Euclidean domain",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0SP/0SPYEFH7H4.md",
  "status": null,
  "tag": "0SPYEFH7H4",
  "uri": "m2://0SPYEFH7H4",
  "url": "https://math2.tamnd.com/wiki/the-gaussian-integers-form-a-euclidean-domain/",
  "writing": {
    "tag": "0SPYEFH7H4",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}