Proposition. groups of automorphisms of fields [2104]

Let $K / F$ be a field extension.

1. $\mathrm{Aut}(K / F)$ is a group under composition and $\mathrm{Aut}(K / F)$ is a subgroup of $\mathrm{Aut}(K)$.
2. If $L$ is a field and $F \subseteq K \subseteq L$, then$$\mathrm{Aut}(L / K) \leq \mathrm{Aut}(L / F).$$