Definition:Galois Group of Field Extension

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 $G(L/K)$.

Source of Name

This entry was named for Évariste Galois.