[[Field theory MOC]]
# Automorphism of a field extension

Let $L:K$ be a [[field extension]]. An [[automorphism]] $\varphi \in \Aut(L:K)$ of $L:K$ is a [[field automorphism]] of $L$ fixing (the image of) $K$ pointwise, #m/def/field  i.e.
$$
\begin{align*}
\Aut(L:K) = \{ \varphi \in \Aut(L) : \varphi \restriction K= \id_{K} \}
\end{align*}
$$
In the case of a [[Galois extension]], this is denoted $\Gal(L:K)$ and called the [[Galois group]].
An automorphism of a field extension is a special case of an [[Morphism of field extensions]].

## Properties

- [[Bound on the automorphism group of a finite simple extension]]

#
---
#state/tidy | #lang/en | #SemBr