---
tag: 8B57B7EGK4
kind: concept
name: Separably closed field
msc: [12F10]
specializes: [04C5ZBG1WT]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_closure
---

## Summary

A separably closed field is a field with no proper finite [[separable extension|0WCX7NB4N7]].
Equivalently, every [[separable polynomial|0RZ89QWTSJ]] over it of positive degree has a root in it.
Every [[algebraically closed field|0PF6TM8TWT]] is separably closed, and a field is algebraically closed if and only if it is separably closed and [[perfect|VWBVKT794W]].
Separably closed fields have trivial absolute Galois group, and the [[separable closure|JJ91A1V8NQ]] of a field is the separably closed field built from it.

## Explanations

### L1 human

An algebraically closed field contains a root of every polynomial.
A separably closed field only promises a root for polynomials without repeated roots.
In characteristic $0$ the two notions agree.
In characteristic $p$ they differ: the separable closure of $\F_p(t)$ is separably closed, but it has no root of $x^p - t$.
So separably closed fields are as closed as Galois theory can detect.