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  "body": "\n## Definition\n\nLet $L$ be a [[field|0QA8CHH8MG]] and let $H$ be a group of ring automorphisms of $L$.\nThe fixed field of $H$ is\n$$L^H = \\{ a \\in L : \\sigma(a) = a \\text{ for every } \\sigma \\in H \\}.$$\n\n## Remarks\n\nThe set $L^H$ is a [[subfield|07QY5RPFBR]] of $L$, because each $\\sigma \\in H$ respects the field operations.\nFor $L = \\mathbb{C}$ and $H = \\{\\mathrm{id}, \\sigma\\}$, with $\\sigma$ complex conjugation, $L^H = \\mathbb{R}$.\nIf $H$ is finite, then $[L : L^H] = |H|$ and $L/L^H$ is a [[Galois extension|0DS2F7GK05]], by a theorem of Emil Artin.\nMore generally, a group acting on a ring has a fixed subring, the ring of invariants.\n\n## Explanations\n\n### L1 human\n\nThe fixed field is the part of the field that a group of symmetries does not move.\nIn the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], fixed fields turn subgroups into intermediate fields.\n",
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