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:
- $\Gal {L / K} = \leftset {\sigma: L \to \overline K: \sigma}$ is an embedding of $L$ such that $\sigma$ fixes $K$ point-wise$\rightset {}$
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 $\Aut {L / K}$ as a Galois group only 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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Galois group
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Where to begin...
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Galois group