---
tag: 0DS2F7GK05
kind: definition
name: Galois extension
msc: [12F10]
defines: 0QMZAQ6QQ4
uses: [0CBRFQ7DVZ, 0N019F6RE9]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_extension
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Galois_extension
    locator: Characterization of Galois extensions
---

## Definition

An [[algebraic extension|09EYMKMSB9]] $L/K$ is a Galois extension if it is [[normal|0CBRFQ7DVZ]] and [[separable|0N019F6RE9]].

## Remarks

For a finite extension $L/K$, these conditions are equivalent.

1. $L/K$ is Galois.
2. $L$ is the splitting field of a separable polynomial over $K$.
3. $|\operatorname{Aut}(L/K)| = [L : K]$.
4. $K$ is the [[fixed field|0KMJ6919WS]] of $\operatorname{Aut}(L/K)$.

Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$ and every extension of finite fields $\mathbb{F}_{p^n}/\mathbb{F}_p$. The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not Galois.

## Explanations

### L1 human

A Galois extension has as many symmetries over the base field as its degree allows.
For these extensions, the subgroups of the Galois group describe all the intermediate fields.