{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p$ be a prime number and let $n \\ge 1$ and $r \\ge 1$ be integers.\nLet $f \\in (\\Z/p^n\\Z)[x]$ be a [[monic polynomial|0C3QBCR46M]] of degree $r$ whose reduction modulo $p$ is [[irreducible|03FKX7Z5DC]] in $\\F_p[x]$.\nThe *Galois ring* of characteristic $p^n$ and rank $r$ is the [[quotient ring|0076HWHGHQ]]\n$$\\mathrm{GR}(p^n, r) = (\\Z/p^n\\Z)[x] / (f).$$\nHere $\\Z/p^n\\Z$ is the ring of [[integers modulo|03VFZ2WETT]] $p^n$.\nUp to isomorphism, this ring depends only on $p$, $n$ and $r$, and not on the choice of $f$.\n\n## Remarks\n\nTake $p = 2$, $n = 2$, $r = 2$ and $f = x^2 + x + 1$.\nThe polynomial $f$ is monic, and modulo $2$ it has no root in $\\F_2$, so it is irreducible of degree $2$ over $\\F_2$.\nHence $\\mathrm{GR}(4, 2) = (\\Z/4\\Z)[x]/(x^2 + x + 1)$ is a Galois ring with $4^2 = 16$ elements.\nThe ring $(\\Z/4\\Z)[x]/(x^2 + 1)$ is not obtained this way, because $x^2 + 1 \\equiv (x + 1)^2$ modulo $2$ is reducible.\nThe ring $\\mathrm{GR}(p^n, r)$ has $p^{nr}$ elements and characteristic $p^n$.\nIt is a local ring with maximal ideal $p \\, \\mathrm{GR}(p^n, r)$ and residue field the [[finite field|0ZXZ82FTX2]] $\\F_{p^r}$.\nThe special cases are $\\mathrm{GR}(p, r) = \\F_{p^r}$ and $\\mathrm{GR}(p^n, 1) = \\Z/p^n\\Z$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "H9S7RMBMFD",
    "kind": "definition",
    "msc": [
      "13M05",
      "11T30"
    ],
    "name": "Galois ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Finite_ring"
      }
    ],
    "tag": "8Y0M6X8Q2A",
    "uses": [
      "03VFZ2WETT",
      "0C3QBCR46M",
      "03FKX7Z5DC",
      "0076HWHGHQ",
      "0ZXZ82FTX2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/8Y0M6X8Q2A/",
  "links": [
    {
      "field": "defines",
      "target": "H9S7RMBMFD"
    },
    {
      "field": "uses",
      "target": "03VFZ2WETT"
    },
    {
      "field": "uses",
      "target": "0C3QBCR46M"
    },
    {
      "field": "uses",
      "target": "03FKX7Z5DC"
    },
    {
      "field": "uses",
      "target": "0076HWHGHQ"
    },
    {
      "field": "uses",
      "target": "0ZXZ82FTX2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/8Y0M6X8Q2A.md",
  "msc": [
    "13M05",
    "11T30"
  ],
  "name": "Galois ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8Y0/8Y0M6X8Q2A.md",
  "status": null,
  "tag": "8Y0M6X8Q2A",
  "uri": "m2://8Y0M6X8Q2A",
  "url": "https://math2.tamnd.com/wiki/galois-ring-definition/",
  "writing": {
    "tag": "8Y0M6X8Q2A",
    "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
      }
    ]
  }
}