---
tag: 0N019F6RE9
kind: definition
name: Separable extension
msc: [12F10]
defines: 0WCX7NB4N7
uses: [09EYMKMSB9, 0GK1WB9R98, 03J8BS7FMV]
sources:
  - url: https://en.wikipedia.org/wiki/Separable_extension
    locator: Separable elements and separable extensions
---

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
An element $a \in L$ is separable over $K$ if its [[minimal polynomial|0GK1WB9R98]] over $K$ is a [[separable polynomial|03J8BS7FMV]].
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.