Definition:Galois Group of Field Extension

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

Then the set:


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

forms a group under composition.

$\operatorname{Gal} \left({L / K}\right)$ 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:


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

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

Also known as
The Galois group of $L / K$ is also known as its automorphism group and denoted $\operatorname{Aut}(L/K)$. Some authors refer to $\operatorname{Aut}(L/K)$ only as a Galois group when $L/K$ is a Galois extension.

Also see

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