---
tag: PWQ031W07T
kind: definition
name: Cyclic extension
msc: [12F10]
defines: VFE5B71E7Y
uses: [0DS2F7GK05, 0XVR72M51M, 0B85Q41RP6]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_extension
---

## Definition

Let $L/K$ be a [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $\Gal(L/K)$.
The extension $L/K$ is *cyclic* if $\Gal(L/K)$ is a [[cyclic group|0B85Q41RP6]].
It is *cyclic of degree $n$* if moreover $\Gal(L/K)$ has order $n$.

## Remarks

Example: let $p$ be a prime, $q$ a power of $p$, and $n \ge 1$.
The extension $\F_{q^n}/\F_q$ of finite fields is cyclic of degree $n$, generated by the Frobenius map $x \mapsto x^q$.
Non-example: $\Q(\sqrt{2}, \sqrt{3})/\Q$ is Galois with group $\Z/2 \times \Z/2$.
Every element of this group has order at most $2$, while the group has order $4$, so it is not cyclic.
Every cyclic extension is an [[abelian extension|EQDEXBWGJH]].