{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p$ be a prime number and let $\\Q_p$ be the field of [[p-adic numbers|BT6T702C28]], with the [[p-adic absolute value|WGN82TCEF2]] $\\abs{\\cdot}_p$.\nA *$p$-adic integer* is an element $x \\in \\Q_p$ with $\\abs{x}_p \\le 1$.\nThe set of $p$-adic integers is denoted $\\Z_p$.\n\n## Remarks\n\nBy the strong triangle inequality, $\\Z_p$ is a subring of $\\Q_p$.\nIts units are the $x$ with $\\abs{x}_p = 1$, and its only nonzero prime ideal is $p \\Z_p$.\nEquivalently, $\\Z_p$ is the set of sums $\\sum_{n \\ge 0} a_n p^n$ with digits $a_n \\in \\set{0, \\ldots, p - 1}$.\nEquivalently, $\\Z_p$ is the inverse limit of the rings $\\Z / p^n \\Z$ under reduction maps.\nExample: $1/2 \\in \\Z_3$, because $3$ does not divide $2$, so $\\abs{1/2}_3 = 1$.\nNon-example: $1/3 \\notin \\Z_3$, because $\\abs{1/3}_3 = 3 \u003e 1$.\nA rational number lies in $\\Z_p$ exactly when its reduced denominator is not divisible by $p$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "V6TJAFZ4MD",
    "kind": "definition",
    "msc": [
      "12J25",
      "11S99"
    ],
    "name": "p-adic integer",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/P-adic_number"
      }
    ],
    "tag": "GBN9F8FAQ3",
    "uses": [
      "BT6T702C28",
      "WGN82TCEF2",
      "MGM3CNJW9V"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/GBN9F8FAQ3/",
  "links": [
    {
      "field": "defines",
      "target": "V6TJAFZ4MD"
    },
    {
      "field": "uses",
      "target": "BT6T702C28"
    },
    {
      "field": "uses",
      "target": "WGN82TCEF2"
    },
    {
      "field": "uses",
      "target": "MGM3CNJW9V"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GBN9F8FAQ3.md",
  "msc": [
    "12J25",
    "11S99"
  ],
  "name": "p-adic integer",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GBN/GBN9F8FAQ3.md",
  "status": null,
  "tag": "GBN9F8FAQ3",
  "uri": "m2://GBN9F8FAQ3",
  "url": "https://math2.tamnd.com/wiki/p-adic-integer-definition/",
  "writing": {
    "tag": "GBN9F8FAQ3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}