{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0QRCFZKXH8"
    }
  ],
  "body": "\n## Statement\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] and let $p \\in R$ be a [[prime element|0MNHXBSX3M]].\nThen $p$ is [[irreducible|052SDEPZH7]].\n\n## Remarks\n\nThe proof: let $p = a b$. Then $p \\mid a b$, so $p \\mid a$, say. Write $a = p c$. Then $p = p c b$, and [[cancellation|0GJM24EXP9]] gives $c b = 1$, so $b$ is a unit.\nThe converse is false in general: in $\\mathbb{Z}[\\sqrt{-5}]$, [[$2$ is irreducible but not prime|0CHE3J7QMX]].\nIn a [[unique factorization domain|06YMXJMXYS]], [[irreducible elements are prime|0QRCFZKXH8]].\n\n## Explanations\n\n### L1 human\n\nSuppose that a prime $p$ factors as $a b$.\nThen $p$ divides one factor, so that factor is at least as large as $p$ for divisibility.\nThe other factor must then be a unit.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "13A05"
    ],
    "name": "Prime elements are irreducible in an integral domain",
    "role": "proposition",
    "sources": [
      {
        "locator": "Irreducible elements",
        "url": "https://en.wikipedia.org/wiki/Prime_element"
      }
    ],
    "tag": "0F0K8DB9TD",
    "uses": [
      "0MNHXBSX3M",
      "052SDEPZH7",
      "09S3PFBEPM"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0F0K8DB9TD/",
  "links": [
    {
      "field": "uses",
      "target": "0MNHXBSX3M"
    },
    {
      "field": "uses",
      "target": "052SDEPZH7"
    },
    {
      "field": "uses",
      "target": "09S3PFBEPM"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0F0K8DB9TD.md",
  "msc": [
    "13A05"
  ],
  "name": "Prime elements are irreducible in an integral domain",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0F0/0F0K8DB9TD.md",
  "status": null,
  "tag": "0F0K8DB9TD",
  "uri": "m2://0F0K8DB9TD",
  "url": "https://math2.tamnd.com/wiki/prime-elements-are-irreducible-in-an-integral-domain/",
  "writing": {
    "tag": "0F0K8DB9TD",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}