Definition:Normal Extension

Also see

 * Equivalence of Definitions of Normal Extension
 * Definition:Separable Extension
 * Definition:Galois Extension

Note
The second definition captures the core concept of a normal extension as a field extension in which the embeddings are actually field automorphisms.

This, in turn, endows $\operatorname{Gal} \left({L / K}\right)$ with a group structure since we can now compose embeddings.

However, in developing the theory, the first definition tends to be more useful.