---
tag: EX34J8PP61
kind: definition
name: Brauer group
msc: [16K50, 12G05]
defines: BEW48TA9WZ
uses: [0QA8CHH8MG, 0FHVPRAE60, F0HQXQ0HCD, 071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Brauer_group
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
A central simple algebra over $K$ is a finite-dimensional associative $K$-algebra $A \ne 0$ whose center is $K$ and whose only two-sided ideals are $0$ and $A$.
For a $K$-algebra $A$, let $M_m(A)$ be the [[matrix ring|0FHVPRAE60]] of $m \times m$ matrices over $A$.
Two central simple algebras $A$ and $B$ over $K$ are *Brauer equivalent* if $M_m(A) \cong M_n(B)$ as $K$-algebras for some integers $m, n \ge 1$.
The *Brauer group* $\mathrm{Br}(K)$ is the set of Brauer equivalence classes $[A]$, with the operation
$$[A] \cdot [B] = [A \otimes_K B],$$
the [[tensor product|F0HQXQ0HCD]] over $K$.

## Remarks

The operation is well defined, and $\mathrm{Br}(K)$ is an [[abelian group|071DDFVG4J]] with identity $[K]$ and inverse $[A]^{-1} = [A^{\mathrm{op}}]$, the opposite algebra; this rests on $A \otimes_K A^{\mathrm{op}} \cong M_d(K)$ with $d = \dim_K A$.
By Wedderburn's theorem each central simple algebra is $M_n(D)$ for a central division algebra $D$ unique up to isomorphism, so each class contains exactly one such $D$.
Example: $\mathrm{Br}(\R)$ has two elements, $[\R]$ and $[\mathbb{H}]$, by the [[Frobenius theorem|QDJ7PQE9SW]].
Example: $\mathrm{Br}(K) = 0$ for $K$ algebraically closed and for $K$ finite.
Non-example: the $K$-algebra $K \times K$ has no class in $\mathrm{Br}(K)$, since its center is $K \times K$, not $K$, and $K \times \set{0}$ is a two-sided ideal.
There is an isomorphism $\mathrm{Br}(K) \cong H^2(K, (K^s)^\times)$ with [[Galois cohomology|9MPCZ3EAJM]], where $K^s$ is a separable closure of $K$.