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  "body": "\n## Definition\n\nAn [[algebraic extension|09EYMKMSB9]] $L/K$ is a Galois extension if it is [[normal|0CBRFQ7DVZ]] and [[separable|0N019F6RE9]].\n\n## Remarks\n\nFor a finite extension $L/K$, these conditions are equivalent.\n\n1. $L/K$ is Galois.\n2. $L$ is the splitting field of a separable polynomial over $K$.\n3. $|\\operatorname{Aut}(L/K)| = [L : K]$.\n4. $K$ is the [[fixed field|0KMJ6919WS]] of $\\operatorname{Aut}(L/K)$.\n\nExamples: $\\mathbb{C}/\\mathbb{R}$, $\\mathbb{Q}(\\sqrt{2}, \\sqrt{3})/\\mathbb{Q}$ and every extension of finite fields $\\mathbb{F}_{p^n}/\\mathbb{F}_p$. The extension $\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q}$ is not Galois.\n\n## Explanations\n\n### L1 human\n\nA Galois extension has as many symmetries over the base field as its degree allows.\nFor these extensions, the subgroups of the Galois group describe all the intermediate fields.\n",
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