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 $L$.

Alternatively, we can define a normal extension in terms of the embeddings of $L$ in an algebraic closure of $K$. Let $\text{Gal}(L/K)$ 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 $\sigma\in \text{Gal}(L/K)$.

Note
The second definition captures the core concept of a normal extension as a field extension in which the embeddings are actually automorphisms. This, in turn, endows $\text{Gal}(L/K)$ with a group structure since we can now compose embeddings. However, in developing the theory, the first definition tends to be more useful.