Definition:Galois Group of Field Extension

Definition
Let $L/K$ be a Galois extension.

Then the set:


 * $\operatorname{Gal}(L/K) = \left\{{\sigma: L \to L: \sigma}\right.$ is an automorphism of $L$ such that $\sigma$ fixes $K$ point-wise$\left.\right\}$

forms a group under composition.

$\operatorname{Gal}(L/K)$ is called the Galois group of $L/K$.

Alternative Definition
More generally, we can abandon the condition that $L/K$ be Galois if we choose an algebraic closure $\overline{K}$ such that $L \subseteq \overline{K}$ and define:


 * $\operatorname{Gal}(L/K) = \left\{{\sigma: L \to \overline{K}: \sigma}\right.$ is an embedding of $L$ such that $\sigma$ fixes $K$ point-wise$\left.\right\}$

This set will form a group iff $L/K$ is normal.

Also see

 * Galois Group is a Group for a proof that the above statements hold, and that these definitions are justified.