{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0M893X7YY7"
    },
    {
      "field": "uses",
      "target": "0VG02M1FEZ"
    },
    {
      "field": "uses",
      "target": "0ZN39QDTP9"
    }
  ],
  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$.\nThen the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$ is a field if and only if $f$ is [[irreducible|00X2A9G9W3]] over $K$.\n\n## Remarks\n\nThe proof: $K[x]$ is a [[principal ideal domain|0RZYDVZ475]]. So $(f)$ is maximal if and only if $f$ is irreducible, by the [[theorem on PIDs|095ZHQ7WCJ]]. Then [[$R/M$ is a field iff $M$ is maximal|0WD3SE7S1X]].\nFor example, $\\mathbb{R}[x]/(x^2 + 1) \\cong \\mathbb{C}$ and $\\mathbb{Q}[x]/(x^2 - 2) \\cong \\mathbb{Q}(\\sqrt{2})$.\nThe field $\\mathbb{F}_2[x]/(x^2 + x + 1)$ has $4$ elements.\nIf $f$ is irreducible of degree $n$, then $K[x]/(f)$ is an extension of $K$ of degree $n$.\nThe class of $x$ in $K[x]/(f)$ is a root of $f$.\n\n## Explanations\n\n### L1 human\n\nYou can build new fields from polynomials.\nTake an irreducible polynomial $f$ and declare $f(x) = 0$.\nThe result is a field in which $f$ has a root, as the complex numbers come from $x^2 + 1 = 0$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "12E05"
    ],
    "name": "K[x]/(f) is a field iff f is irreducible",
    "role": "theorem",
    "sources": [
      {
        "locator": "Field extension",
        "url": "https://en.wikipedia.org/wiki/Irreducible_polynomial"
      }
    ],
    "tag": "0AWMPX35CW",
    "uses": [
      "095ZHQ7WCJ",
      "0WD3SE7S1X",
      "0DQG39K2NH",
      "00X2A9G9W3"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0AWMPX35CW/",
  "links": [
    {
      "field": "uses",
      "target": "095ZHQ7WCJ"
    },
    {
      "field": "uses",
      "target": "0WD3SE7S1X"
    },
    {
      "field": "uses",
      "target": "0DQG39K2NH"
    },
    {
      "field": "uses",
      "target": "00X2A9G9W3"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0AWMPX35CW.md",
  "msc": [
    "12E05"
  ],
  "name": "K[x]/(f) is a field iff f is irreducible",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0AW/0AWMPX35CW.md",
  "status": null,
  "tag": "0AWMPX35CW",
  "uri": "m2://0AWMPX35CW",
  "url": "https://math2.tamnd.com/wiki/k-x-f-is-a-field-iff-f-is-irreducible/",
  "writing": {
    "tag": "0AWMPX35CW",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}