Fixed field

Let \(L/k\) be a field extension and let \(H \leq \Aut_{\Fieldsover{k}}(L)\) be a subgroup of the automorphisms of \(L\) fixing \(k\). The fixed field of \(H\) is \[ L^H \coloneqq \theset{ \ell \in L \st \sigma(\ell) = \ell \text{ for all } \sigma\in H }. \]