Field automorphism group
= Field automorphism group
{title2=$\operatorname{Aut}_K(L)$}
The field automorphism group fixing $K$ consists of all <field automorphisms> of $L$ fixing $K$ pointwise, with composition. It is defined without assuming that $L/K$ is Galois. For $L=K(\alpha)$ with algebraic degree $d$, the action on the roots of the <minimal polynomial> inside $L$ is faithful, so this group embeds in $S_d$ and is finite.