Normal extension

Definition

Let $L/K$ be an algebraic extension. The extension $L/K$ is normal if every irreducible polynomial $f \in K[x]$ that has a root in $L$ splits over $L$.

Remarks

A finite extension $L/K$ is normal if and only if $L$ is the splitting field of some polynomial over $K$. Every extension of degree $2$ is normal. For example, $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is normal. The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal: $x^3 - 2$ has a root in it, but its other two roots are not real. Normality is not transitive: $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ are normal, but $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ is not.

Explanations

L1 human

A normal extension contains all the roots of a polynomial as soon as it contains one of them. It has no lopsided gaps, like a field that contains one cube root of $2$ but not the others.