Normal extension - Imported Knowlpedia Content

Let $K/F$ be an algebraic field extension.

Definition (normal extension). The extension $K/F$ is normal if for every irreducible polynomial $p(x)\in F[x]$ , the condition “ $p$ has a root in $K$ ” implies “ $p$ splits into linear factors in $K[x]$ ”.

Equivalent characterizations

There are several standard equivalent characterizations (assuming $K/F$ is algebraic):

Fix an algebraic closure $\overline F$ containing $K$ . Then $K/F$ is normal iff every $F$ -embedding $\sigma:K\hookrightarrow \overline F$ satisfies $\sigma(K)=K$ .
$K/F$ is normal iff $K$ is the splitting field over $F$ of some family of polynomials in $F[x]$ . If $K/F$ is finite, it suffices to take a single polynomial.

Remarks

Splitting fields are normal (see normality of splitting fields). A normal extension need not be separable; when it is both normal and separable, it is Galois.

Examples

$\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is normal: it is the splitting field of $x^2-2$ .
$\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal: the irreducible polynomial $x^3-2$ has one root $\sqrt[3]{2}$ in $\mathbb{Q}(\sqrt[3]{2})$ , but its other roots $\zeta_3\sqrt[3]{2}$ and $\zeta_3^2\sqrt[3]{2}$ are not in that field.
For finite fields, $\mathbb{F}_{p^n}/\mathbb{F}_p$ is normal: it is the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ .