---
tag: VFE5B71E7Y
kind: concept
name: Cyclic extension
msc: [12F10]
specializes: [EQDEXBWGJH]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_extension
---

## Summary

A cyclic extension is a [[Galois extension|0DS2F7GK05]] whose [[Galois group|0XVR72M51M]] is [[cyclic|0B85Q41RP6]].
Cyclic extensions are the building blocks of radical extensions.
If a field contains $n$ distinct $n$-th roots of unity, adjoining an $n$-th root of one of its elements gives a cyclic extension.
[[Kummer theory|XZQENYV15Y]] and [[Artin–Schreier theory|7YNMNRGH3T]] describe cyclic extensions of a given degree.

## Explanations

### L1 human

A cyclic extension has one basic symmetry, and every other symmetry is a repeat of it.
Every finite extension of finite fields is cyclic, generated by the map $x \mapsto x^q$, where $q$ is the size of the smaller field.
For example, $\Q(\sqrt{5})$ over $\Q$ is cyclic of degree $2$, generated by $\sqrt{5} \mapsto -\sqrt{5}$.
Every cyclic extension is abelian, but $\Q(\sqrt{2}, \sqrt{3})$ over $\Q$ is abelian and not cyclic.