Definition:Galois Group of Field Extension/Also defined as

Galois Group: Also defined as

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.

Sources

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• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Galois group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Galois group