Automorphism group of a field extension

Let \(L/k\) be a field extension. The automorphism group of \(L/k\) is the group, under composition, \[ \Aut(L/k) \coloneqq \theset{\sigma\colon L \to L \text{ a field automorphism} \suchthat \restrictionof{\sigma}{k} = \id_k}. \]