{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "069A3BC3EP"
    },
    {
      "field": "uses",
      "target": "0N5QW9K7MH"
    }
  ],
  "body": "\n## Statement\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \\in L$ be algebraic over $K$ with [[minimal polynomial|0GK1WB9R98]] $m_a$.\nThen the [[evaluation map|0125QVNCFQ]] $K[x] \\to L$, $f \\mapsto f(a)$, induces a ring isomorphism\n$$K[x]/(m_a) \\cong K(a).$$\nIn particular, $K(a) = K[a] = \\{ f(a) : f \\in K[x] \\}$.\n\n## Remarks\n\nThe proof: the kernel of the evaluation map is $(m_a)$, by [[the divisibility property|06K7SARZKR]]. The [[first isomorphism theorem|0ZMYX7GMH0]] gives $K[x]/(m_a) \\cong K[a]$.\nSince $m_a$ is irreducible, $K[x]/(m_a)$ is a [[field|0AWMPX35CW]]. So $K[a]$ is a field, and $K[a] = K(a)$.\nIf $a$ is transcendental over $K$, then the evaluation map is injective, and it extends to an isomorphism $K(x) \\cong K(a)$.\n\n## Explanations\n\n### L1 human\n\nAn algebraic element adds nothing to the field beyond the relation given by its minimal polynomial.\nSo $K(a)$ is the same as polynomials in a variable, computed modulo the minimal polynomial.\nFor example, $\\mathbb{Q}(\\sqrt{2}) \\cong \\mathbb{Q}[x]/(x^2 - 2)$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "12F05"
    ],
    "name": "K(a) is isomorphic to K[x] modulo the minimal polynomial",
    "role": "theorem",
    "sources": [
      {
        "locator": "Structure of simple extensions",
        "url": "https://en.wikipedia.org/wiki/Simple_extension"
      }
    ],
    "tag": "0ZN39QDTP9",
    "uses": [
      "0GK1WB9R98",
      "06K7SARZKR",
      "0125QVNCFQ",
      "0ZMYX7GMH0",
      "0AWMPX35CW",
      "0P6SBR7YM2"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0ZN39QDTP9/",
  "links": [
    {
      "field": "uses",
      "target": "0GK1WB9R98"
    },
    {
      "field": "uses",
      "target": "06K7SARZKR"
    },
    {
      "field": "uses",
      "target": "0125QVNCFQ"
    },
    {
      "field": "uses",
      "target": "0ZMYX7GMH0"
    },
    {
      "field": "uses",
      "target": "0AWMPX35CW"
    },
    {
      "field": "uses",
      "target": "0P6SBR7YM2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0ZN39QDTP9.md",
  "msc": [
    "12F05"
  ],
  "name": "K(a) is isomorphic to K[x] modulo the minimal polynomial",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0ZN/0ZN39QDTP9.md",
  "status": null,
  "tag": "0ZN39QDTP9",
  "uri": "m2://0ZN39QDTP9",
  "url": "https://math2.tamnd.com/wiki/k-a-is-isomorphic-to-k-x-modulo-the-minimal-polynomial/",
  "writing": {
    "tag": "0ZN39QDTP9",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}