---
tag: ZWXW0T3EAC
kind: concept
name: Quaternion algebra
msc: [16K20, 11R52]
specializes: [5HRKJBAY2V]
sources:
  - url: https://en.wikipedia.org/wiki/Quaternion_algebra
same_as:
  wikidata: Q2835967
  wikipedia: Quaternion algebra
---

## Summary

A quaternion algebra over a field $F$ is a [[central simple algebra|5HRKJBAY2V]] over $F$ of dimension $4$.
When the characteristic of $F$ is not $2$, it has a basis $1, i, j, k$ with $i^2 = a$, $j^2 = b$ and $ij = -ji = k$ for some nonzero $a, b \in F$.
Each one is either a division algebra or isomorphic to the algebra of $2 \times 2$ matrices over $F$.
The class of a quaternion algebra in the Brauer group has order at most $2$.
Quaternion algebras are tied to quadratic forms and the Hilbert symbol.

## Explanations

### L1 human

Hamilton's [[quaternions|0GE4DVQKR8]] are numbers $x + yi + zj + wk$ with $i^2 = j^2 = -1$ and $ij = -ji = k$.
A quaternion algebra keeps this pattern but allows other values for $i^2$ and $j^2$ and other fields of coefficients.
With $i^2 = j^2 = -1$ over $\R$ one gets Hamilton's quaternions, a division algebra.
With $i^2 = j^2 = 1$ over $\R$ one gets an algebra isomorphic to the $2 \times 2$ real matrices, which has zero divisors such as $1 + i$.