Normal extension

Definition. Let $L/K$ be an algebraic extension . The extension $L/K$ is normal if every irreducible polynomial $f(x)\in K[x]$ that has at least one root in $L$ actually splits completely into linear factors over $L$.

Equivalently, $L$ is a splitting field of some family of polynomials in $K[x]$. (In many common situations, $L$ is the splitting field of a single polynomial.)

See also. separable extension , Galois extension , normality and splitting fields .

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: $x^3-2$ has a root $\sqrt[3]{2}$ in the field, but does not split there.
3. $\mathbb{F}_{p^n}/\mathbb{F}_p$ is normal: it is the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$.