{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 2$ be an integer.\nThe ring $\\mathbb{Z}/n\\mathbb{Z}$ of integers modulo $n$ is a [[field|0QA8CHH8MG]] if and only if $n$ is a prime number.\n\n## Remarks\n\nIf $n = a b$ with $1 \u003c a, b \u003c n$, then the classes of $a$ and $b$ are nonzero and their product is $0$. So $\\mathbb{Z}/n\\mathbb{Z}$ is not an integral domain.\nIf $n = p$ is prime and $p \\nmid a$, then Bézout's identity gives $a x + p y = 1$ for some integers $x, y$. So $x$ is an inverse of $a$ modulo $p$.\nThe field $\\mathbb{Z}/p\\mathbb{Z}$ is often written $\\mathbb{F}_p$. It is the smallest [[finite field|08RTDHX1SK]] of characteristic $p$.\n\n## Explanations\n\n### L1 human\n\nModulo $5$, every nonzero class has an inverse: $1 \\cdot 1 = 1$, $2 \\cdot 3 = 6 \\equiv 1$ and $4 \\cdot 4 = 16 \\equiv 1$.\nModulo $6$, the class of $2$ has no inverse, because $2 x$ is always even.\nDivision works modulo $n$ exactly when $n$ is prime.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "11A07"
    ],
    "name": "Z/pZ is a field for p prime",
    "role": "theorem",
    "sources": [
      {
        "locator": "Integers modulo m",
        "url": "https://en.wikipedia.org/wiki/Modular_arithmetic"
      }
    ],
    "tag": "0NV6E43P4A",
    "uses": [
      "0QA8CHH8MG",
      "0EBE031B8E"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0NV6E43P4A/",
  "links": [
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "0EBE031B8E"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NV6E43P4A.md",
  "msc": [
    "11A07"
  ],
  "name": "Z/pZ is a field for p prime",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NV/0NV6E43P4A.md",
  "status": null,
  "tag": "0NV6E43P4A",
  "uri": "m2://0NV6E43P4A",
  "url": "https://math2.tamnd.com/wiki/z-pz-is-a-field-for-p-prime/",
  "writing": {
    "tag": "0NV6E43P4A",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}