Definition:Galois Group of Field Extension

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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:

$\Gal {L / K} = \set{\sigma \in \Aut L: \forall k \in K: \map \sigma k = k}$


As a topological group

The notation $\Gal {L / K}$ is also a shorthand for the topological group:

$\struct {\Gal {L / K}, \tau}$

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:

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

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


Also denoted as

The Galois group of $L / K$ can also be denoted $\map G {L / K}$.


Also known as

The Galois group of $L / K$ is also known as its automorphism group and denoted $\Aut {L / K}$.

Some authors refer to $\Aut {L / K}$ only as a Galois group when $L / K$ is a Galois extension.

Some sources use the notation $\map G {L \mid K}$.


Also see


Source of Name

This entry was named for Évariste Galois.


Sources