---
tag: JJ91A1V8NQ
kind: concept
name: Separable closure
msc: [12F10]
sources:
  - url: https://en.wikipedia.org/wiki/Separable_extension
---

## Summary

The separable closure of $K$ in an algebraic extension $L$ is the set of elements of $L$ that are [[separable|0WCX7NB4N7]] over $K$.
It is the largest separable subextension of $L/K$.
The separable closure of a field $K$, written $K^{\mathrm{sep}}$, is the separable closure of $K$ in an [[algebraic closure|0X6AB5P3G1]]; it is a [[separably closed field|8B57B7EGK4]].
Its automorphism group over $K$ is the [[absolute Galois group|PGKWBJDTKX]] of $K$.
For a [[perfect field|VWBVKT794W]] the separable closure is the whole algebraic closure.

## Explanations

### L1 human

Inside a big field, collect all the elements whose minimal polynomials over $K$ have no repeated roots.
These elements form a field, and it is the separable closure.
In characteristic $0$ every polynomial behaves well, so nothing is lost and one gets the algebraic closure.
In characteristic $p$, elements such as a $p$-th root of $t$ over $\F_p(t)$ are left out.
The separable closure is the part of the algebraic closure that Galois theory sees.