Proposition. fixed field [2105]

Let $K / F$ be a field extension.

1. Given $H \leq \mathrm{Aut}(K / F)$, define$$K^{H} = \{ \alpha \in K \mid \sigma(\alpha) = \alpha \text{ for all } \sigma \in H \}.$$Then $K^{H}$ is a subfield of $K$ and $H \leq \mathrm{Aut}(K / K^{H})$. $K^{H}$ is called the fixed field of $H$.
2. For $H_{1} \leq H_{2} \leq \mathrm{Aut}(K / F), K^{H_{1}} \supseteq K^{H_{2}}$.