Separable extension

Definition

Let $L/K$ be an algebraic extension. An element $a \in L$ is separable over $K$ if its minimal polynomial over $K$ is a separable polynomial. The extension $L/K$ is separable if every element of $L$ is separable over $K$.

Remarks

An irreducible polynomial over $K$ is separable if and only if its derivative is not the zero polynomial. So every algebraic extension of a field of characteristic $0$ is separable. Every algebraic extension of a finite field is separable. For a prime $p$ and the field $K = \mathbb{F}_p(t)$ of rational functions, the extension $K(t^{1/p})/K$ is not separable: the minimal polynomial $x^p - t$ equals $(x - t^{1/p})^p$.

Explanations

L1 human

In a separable extension, the roots of each minimal polynomial are all different. Problems arise only in characteristic $p$, where a derivative like that of $x^p - t$ can vanish.