Normal Extensions via Splitting Fields

Theorem.
Let $L/K$ be a finite extension. The following are equivalent:

1. $L/K$ is a normal extension .
2. $L$ is the splitting field over $K$ of a set of polynomials in $K[x]$ (equivalently, of one polynomial).
3. Every $K$-embedding $L \hookrightarrow \overline{K}$ into an algebraic closure has image equal to $L$.

Combined with separability, this characterizes Galois extensions (see separable + normal ⇔ Galois ).

Examples.

1. $\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is normal: it is the splitting field of $x^2-2$.
2. $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal: $x^3-2$ does not split over $\mathbb{Q}(\sqrt[3]{2})$.
3. $\mathbb{F}_{p^n}/\mathbb{F}_p$ is normal: it is the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$.

Related. splitting fields exist and are unique .