---
tag: D6BM80QYNE
kind: concept
name: Cyclic algebra
msc: [16K20, 16S35, 11R52]
specializes: [G21MZQCME9]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_algebra
same_as:
  wikidata: Q56351016
  wikipedia: Cyclic algebra
---

## Summary

A cyclic algebra is a [[crossed product algebra|G21MZQCME9]] for a [[cyclic extension|VFE5B71E7Y]] of fields.
It is described by a generator $\sigma$ of the Galois group and one nonzero element $a$ of the base field.
Every cyclic algebra of degree $2$ is a quaternion algebra, and over local and global fields every central simple algebra is cyclic.
Whether a cyclic algebra splits is decided by the [[field norm|KRNFEZWPY0]]: it splits exactly when $a$ is a norm.

## Explanations

### L1 human

Take a field extension $L/K$ whose symmetries form a cyclic group of order $n$ generated by $\sigma$.
Add one new symbol $u$ to $L$ with two rules.
Moving a number $x$ of $L$ past $u$ turns it into $\sigma(x)$, and $u^n$ equals a fixed number $a$ of $K$.
The result is the cyclic algebra.
For $L = \C$, $K = \R$, $\sigma$ complex conjugation and $a = -1$, this gives Hamilton's quaternions with $u = j$.