\begin{definition}[Definition:Galois Group of Field Extension]
Let $L / K$ be a field extension.
The '''Galois group''' of $L / K$ is the subgroup of the automorphism group of $L$ consisting of field automorphisms that fix $K$ point-wise:
:$\Gal {L / K} = \set {\sigma \in \Aut L: \forall k \in K: \map \sigma k = k}$
\end{definition}