---
tag: NVD7R0T835
kind: concept
name: Splitting field of a central simple algebra
msc: [16K50, 16K20]
sources:
  - url: https://en.wikipedia.org/wiki/Central_simple_algebra
---

## Summary

Let $A$ be a [[central simple algebra|5HRKJBAY2V]] over a field $K$.
A field extension $L$ of $K$ splits $A$ if $A$ becomes a full matrix algebra after extending scalars to $L$.
Every central simple algebra has a splitting field of finite degree over $K$, and even one that is a finite Galois extension of $K$.
Splitting fields link central simple algebras with Galois cohomology: the algebras split by $L$ give the relative Brauer group $\mathrm{Br}(L/K)$.
The word is unrelated to the [[splitting field|0KXQKX9P63]] of a polynomial, apart from the idea of making something decompose.

## Explanations

### L1 human

A central simple algebra may be a division algebra over a small field, yet look like an ordinary matrix algebra over a larger one.
A splitting field is a larger field that makes this happen.
For example, Hamilton's quaternions are a division algebra over $\R$, but over $\C$ they become the algebra of $2 \times 2$ complex matrices, so $\C$ splits them.