---
tag: 5HRKJBAY2V
kind: concept
name: Central simple algebra
msc: [16K20, 16K50]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Central_simple_algebra
same_as:
  wikidata: Q2428439
  wikipedia: Central simple algebra
---

## Summary

A central simple algebra over a field $K$ is a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$ that is a [[simple ring|62M0NC3E2T]].
Its [[center|MVT64B3B6H]] must be exactly $K$.
By Wedderburn's theorem each one is a matrix algebra $M_n(D)$ over a central division algebra $D$.
They are stable under tensor product over $K$, and their classes up to matrix size form the [[Brauer group|BEW48TA9WZ]] of $K$.
They are central in algebraic number theory, in Galois cohomology and in the arithmetic of quadratic forms.

## Explanations

### L1 human

The basic example is the algebra of $n \times n$ matrices over a field $K$.
It is simple, since its only two-sided ideals are $0$ and itself, and its center consists of the scalar matrices, which form a copy of $K$.
Central simple algebras are the algebras that look like matrix algebras after the field is enlarged.
For example, Hamilton's [[quaternions|0GE4DVQKR8]] are central simple over $\R$, and over $\C$ they become the $2 \times 2$ complex matrices.