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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]], and let $\\Aut(L/K)$ be the group of field automorphisms of $L$ that fix every element of $K$.\n*Galois theory* is the study of field extensions $L/K$ through the groups $\\Aut(L/K)$ and the [[Galois connection between subgroups and subfields|R3625YKQC0]].\nIts basic objects are:\n\n1. the [[Galois extensions|0DS2F7GK05]], that is, the algebraic extensions that are normal and separable;\n2. their [[Galois groups|0XVR72M51M]] $\\Gal(L/K) = \\Aut(L/K)$;\n3. the maps $H \\mapsto L^H$, from subgroups of $\\Gal(L/K)$ to their [[fixed fields|0KMJ6919WS]], and $E \\mapsto \\Gal(L/E)$, from [[intermediate fields|02NBKVQFRQ]] to subgroups.\n\n## Remarks\n\nExample: for $L = \\Q(i)$ over $\\Q$, the group $\\Gal(L/\\Q)$ has two elements, the identity and complex conjugation, and the fixed field of the whole group is $\\Q$.\nNon-example of a Galois extension: $\\Q(\\sqrt[3]{2})/\\Q$ is separable but not normal, since $x^3 - 2$ has only one root in it.\nIts automorphism group is trivial, so the base field $\\Q$ is not the fixed field of any subgroup.\nThe Galois group of a separable polynomial $f \\in K[x]$ is the Galois group of a splitting field of $f$ over $K$.\n",
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