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  "body": "\n## Statement\n\nLet $L/K$ be a [[finite extension|0NRYPM9Y27]].\nThen the group $\\operatorname{Aut}(L/K)$ of [[automorphisms of $L/K$|0XVR72M51M]] is finite, and\n$$|\\operatorname{Aut}(L/K)| \\le [L : K].$$\n\n## Remarks\n\nFor a simple extension $L = K(a)$, an automorphism is determined by $\\sigma(a)$, which is a root in $L$ of the minimal polynomial of $a$. There are at most $[L : K]$ such roots.\nEquality holds if and only if $L/K$ is a [[Galois extension|0DS2F7GK05]].\nFor $\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q}$, the inequality is strict: $1 \u003c 3$.\n\n## Explanations\n\n### L1 human\n\nA finite extension has only finitely many symmetries over the base field, and at most as many as its degree.\nWhen there are exactly as many as the degree, the extension is Galois.\n",
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        "locator": "Characterization of Galois extensions",
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