---
tag: EQDEXBWGJH
kind: concept
name: Abelian extension
msc: [12F10, 11R20]
specializes: [0QMZAQ6QQ4]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_extension
same_as:
  wikidata: Q318705
  wikipedia: Abelian extension
---

## Summary

An abelian extension is a [[Galois extension|0DS2F7GK05]] whose [[Galois group|0XVR72M51M]] is abelian.
By the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], every intermediate field of a finite abelian extension is again Galois over the base.
[[Cyclic extensions|VFE5B71E7Y]] and [[cyclotomic extensions|VD6VC3ZMMY]] of $\Q$ are abelian.
Class field theory describes the abelian extensions of number fields and local fields.

## Explanations

### L1 human

An extension is abelian when its symmetries commute with each other.
Then the order in which two symmetries are applied does not matter.
For example, $\Q(\sqrt{2}, \sqrt{3})$ over $\Q$ has four symmetries, each changing the signs of $\sqrt{2}$ and of $\sqrt{3}$ independently.
Changing the two signs in either order gives the same result, so the extension is abelian.