{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a field and let $A$ be a [[central simple algebra|5HRKJBAY2V]] over $K$.\nLet $B$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$ that is a [[simple ring|62M0NC3E2T]].\nLet $f, g \\colon B \\to A$ be homomorphisms of $K$-algebras.\nThen there is an invertible element $u \\in A$ such that $g(x) = u f(x) u^{-1}$ for all $x \\in B$.\nIn particular, every automorphism of the $K$-algebra $A$ is an inner automorphism, that is, a conjugation $x \\mapsto u x u^{-1}$.\n\n## Remarks\n\nThe algebra $B \\otimes_K A^{\\mathrm{op}}$ is simple and finite-dimensional, so it has only one simple module up to isomorphism, and its modules are determined by their dimension.\nThe maps $f$ and $g$ make $A$ into two $B \\otimes_K A^{\\mathrm{op}}$-modules, by $(b \\otimes a) \\cdot x = f(b) x a$ and $(b \\otimes a) \\cdot x = g(b) x a$, of the same dimension.\nAn isomorphism $\\varphi$ from the first to the second commutes with right multiplication by $A$, so $\\varphi(x) = u x$ with $u = \\varphi(1)$ invertible, and compatibility with $B$ gives $u f(b) = g(b) u$.\nExample: every automorphism of $M_n(K)$ as a $K$-algebra is conjugation $X \\mapsto P X P^{-1}$ by an invertible matrix $P$.\nExample: any two elements of $\\mathbb{H}$ with the same minimal polynomial $x^2 + 1$ over $\\R$, such as $i$ and $j$, are conjugate in $\\mathbb{H}$.\n\n## Explanations\n\n### L1 human\n\nInside a central simple algebra, two copies of the same simple algebra always differ by a change of coordinates.\nThat change of coordinates is conjugation by an invertible element.\nFor matrices this says that every symmetry of the matrix algebra that respects scalars comes from a change of basis.\n\n### L2 human\n\nThe theorem is used constantly in the theory of central simple algebras.\nIt shows that $\\Aut_K(M_n(K)) \\cong \\mathrm{PGL}_n(K)$, which identifies central simple algebras split by a Galois extension $L$ with classes in $H^1(\\Gal(L/K), \\mathrm{PGL}_n(L))$.\nApplied to a subfield $L$ of $A$ and its Galois conjugates, it is a key step in building the crossed product description of central simple algebras.\nTogether with the double centralizer theorem it controls how subfields sit inside central simple algebras.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "5HRKJBAY2V"
    ],
    "kind": "statement",
    "msc": [
      "16K20",
      "16K50"
    ],
    "name": "Skolem–Noether theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3001796",
      "wikipedia": "Skolem–Noether theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Skolem%E2%80%93Noether_theorem"
      }
    ],
    "tag": "V4PEVFF4F4",
    "uses": [
      "5HRKJBAY2V",
      "62M0NC3E2T",
      "3CTW1P4HBP"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/V4PEVFF4F4/",
  "links": [
    {
      "field": "about",
      "target": "5HRKJBAY2V"
    },
    {
      "field": "uses",
      "target": "5HRKJBAY2V"
    },
    {
      "field": "uses",
      "target": "62M0NC3E2T"
    },
    {
      "field": "uses",
      "target": "3CTW1P4HBP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/V4PEVFF4F4.md",
  "msc": [
    "16K20",
    "16K50"
  ],
  "name": "Skolem–Noether theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V4P/V4PEVFF4F4.md",
  "status": null,
  "tag": "V4PEVFF4F4",
  "uri": "m2://V4PEVFF4F4",
  "url": "https://tamnd.github.io/math2/wiki/skolem-noether-theorem/",
  "writing": {
    "tag": "V4PEVFF4F4",
    "kind": "statement",
    "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
      }
    ]
  }
}