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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nAn automorphism of $L/K$ is a ring isomorphism $\\sigma \\colon L \\to L$ with $\\sigma(c) = c$ for every $c \\in K$.\nThese automorphisms form a [[group|G5S1M9]] $\\operatorname{Aut}(L/K)$ under composition.\nIf $L/K$ is a [[Galois extension|0DS2F7GK05]], this group is the Galois group of $L/K$, written $\\operatorname{Gal}(L/K)$.\n\n## Remarks\n\nSome authors call $\\operatorname{Aut}(L/K)$ the Galois group for every extension $L/K$.\nThe group $\\operatorname{Gal}(\\mathbb{C}/\\mathbb{R})$ has two elements: the identity and complex conjugation.\nThe group $\\operatorname{Aut}(\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q})$ is trivial, because an automorphism must send $\\sqrt[3]{2}$ to a real cube root of $2$.\nFor a finite extension, $|\\operatorname{Aut}(L/K)| \\le [L : K]$, by the [[bound on the order|0QM98DNWVW]].\n\n## Explanations\n\n### L1 human\n\nThe Galois group collects the symmetries of a bigger field that leave the smaller field unchanged.\nEach symmetry permutes the roots of every polynomial over the smaller field.\n",
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