Definition:Normal Extension

Definition
A field extension $L/K$ is a normal extension if for every irreducible polynomial $f\in K[x]$ with at least one root in $L$, $f$ splits completely in $L$.

Alternative Definition
Alternatively, we can define a normal extension in terms of the embeddings of $L$ in an algebraic closure of $K$.

Let $\operatorname{Gal} \left({L / K}\right)$ denote the set of embeddings of $L$ in $\overline{K}$ which fix $K$ pointwise.

Then we say $L/K$ is a normal extension if $\sigma(L) = L$ for each $\sigma \in \operatorname{Gal}(L/K)$.

Also see

 * Equivalence of Definitions of Normal 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.