Normal extensions and splitting fields

Let $K\subseteq L$ be an algebraic field extension . The following are equivalent:

1. $L/K$ is a normal extension .
2. For every irreducible $f(x)\in K[x]$ having at least one root in $L$, the polynomial $f$ splits completely into linear factors over $L$.
3. There exists a set $S\subseteq K[x]$ such that $L$ is the splitting field of $S$ over $K$. If $L/K$ is finite, one may take $S=\{f\}$ for a single polynomial $f\in K[x]$.

In particular, finite normal extensions are exactly finite splitting fields (up to $K$-isomorphism, cf. uniqueness of splitting fields ).

Examples

1. Quadratic extensions are normal.
   $L=\mathbb{Q}(\sqrt2)$ is the splitting field of $x^2-2$ over $\mathbb{Q}$, hence $L/\mathbb{Q}$ is normal.

2. A non-normal cubic.
   $L=\mathbb{Q}(\sqrt[3]{2})$ contains one root of $x^3-2$ but not the complex roots $\sqrt[3]{2}\zeta_3$, $\sqrt[3]{2}\zeta_3^2$. Thus $x^3-2$ does not split over $L$, so $L/\mathbb{Q}$ is not normal.

3. Finite fields as splitting fields.
   For $L=\mathbb{F}_{p^n}$ and $K=\mathbb{F}_p$, the polynomial $x^{p^n}-x\in K[x]$ splits over $L$ with roots exactly the elements of $L$. Hence $L/K$ is normal (and in fact Galois ).