# Definition:Fixed Field

## Definition

Let $F$ be a field.

Let $G \le \Aut F$ be a subgroup of the automorphism group of $F$.

The fixed field of $G$ is the set:

$\Fix G = \set {f \in F : \forall \sigma \in G : \map \sigma f = f}$

## Also denoted as

The fixed field of $G$ can also be denoted $F_G$ or $\map {\operatorname {Fix}_F} G$ to emphasize that it is contained in $F$.