---
tag: NNXSP5KKAC
kind: concept
name: Degree of a central simple algebra
msc: [16K20, 16K50]
sources:
  - url: https://en.wikipedia.org/wiki/Central_simple_algebra
---

## Summary

The dimension of a [[central simple algebra|5HRKJBAY2V]] over its center is always a perfect square $n^2$.
The number $n$ is the degree of the algebra.
It is the size of the matrices that the algebra becomes over an algebraically closed field.
It bounds the index and the period of the algebra.

## Explanations

### L1 human

The algebra of $n \times n$ matrices over a field has dimension $n^2$, and its degree is $n$.
Every central simple algebra becomes such a matrix algebra over a large enough field, so its dimension is also a square.
Hamilton's quaternions have dimension $4$ over $\R$, so their degree is $2$; over $\C$ they become the $2 \times 2$ complex matrices.