Normal extension

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 $K[x]$”.

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.

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.

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: 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.
3. For finite fields, $\mathbb{F}_{p^n}/\mathbb{F}_p$ is normal: it is the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$.