\begin{definition}[Definition:Fixed Field]
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:
:$\Fix G = \set {f \in F : \forall \sigma \in G : \map \sigma f = f}$
\end{definition}