Normal extensions and splitting fields - Imported Knowlpedia Content

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

$L/K$ is a normal extension.
For every irreducible $f(x)\in K[x]$ having at least one root in $L$ , the polynomial $f$ splits completely into linear factors over $L$ .
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 $f\in K[x]$ .

Remarks

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

Examples

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.

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.

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).