# Definition:Galois Group of Field Extension

## Contents

## Definition

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:

- $\operatorname{Gal} \left({L / K}\right) = \left\{ \sigma \in \operatorname{Aut}(L) : \forall k \in K : \sigma (k) = k \right\}$

### As a topological group

The notation $\operatorname{Gal} \left({L / K}\right)$ is also a shorthand for the topological group

- $(\operatorname{Gal} \left({L / K}\right), \tau)$

where $\tau$ is the Krull topology.

### 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 if and only if $L / K$ 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. It is perfectly possible to use a straight line, as in $G(L \mid K)$.

## Also denoted as

The **Galois group** of $L/K$ can also be denoted $G(L/K)$.

## Also see

## Source of Name

This entry was named for Évariste Galois.

## Sources

- 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Where to begin...