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  "body": "\n## Statement\n\nLet $L/K$ be a finite [[Galois extension|0DS2F7GK05]].\nThen\n$$|\\operatorname{Gal}(L/K)| = [L : K].$$\n\n## Remarks\n\nConversely, a finite extension with $|\\operatorname{Aut}(L/K)| = [L : K]$ is Galois. So the [[general bound|0QM98DNWVW]] is an equality exactly for Galois extensions.\nFor $\\mathbb{Q}(\\sqrt{2}, \\sqrt{3})/\\mathbb{Q}$, the Galois group has order $4$: an automorphism can change the signs of $\\sqrt{2}$ and of $\\sqrt{3}$ independently.\nFor $\\mathbb{F}_{p^n}/\\mathbb{F}_p$, the Galois group is cyclic of order $n$, generated by the [[Frobenius automorphism|0VP4TBVJ19]].\n\n## Explanations\n\n### L1 human\n\nFor a Galois extension, counting symmetries is the same as computing the degree.\nThis links the size of a group to the size of a field extension.\n",
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