# Brauer group

## 1 Algebraic view

Let $K$ be a field. The $\operatorname{Br}(K)$ of $K$ is the set of all equivalence classes of central simple algebras over $K$, where two central simple algebras $A$ and $B$ are equivalent if there exists a division ring $D$ over $K$ and natural numbers $n,m$ such that $A$ (resp. $B$) is isomorphic to the ring of $n\times n$ (resp. $m\times m$) matrices with coefficients in $D$.

The group operation in $\operatorname{Br}(K)$ is given by tensor product: for any two central simple algebras $A,B$ over $K$, their product in $\operatorname{Br}(K)$ is the central simple algebra $A\otimes_{K}B$. The identity element in $\operatorname{Br}(K)$ is the class of $K$ itself, and the inverse of a central simple algebra $A$ is the opposite algebra $A^{\operatorname{opp}}$ defined by reversing the order of the multiplication operation of $A$.

## 2 Cohomological view

The Brauer group of $K$ is naturally isomorphic to the second Galois cohomology group $H^{2}({\operatorname{Gal}}(K^{\operatorname{sep}}/K),(K^{\operatorname{sep}})^% {\times})$. See http://www.math.harvard.edu/ elkies/M250.01/index.htmlhttp://www.math.harvard.edu/ elkies/M250.01/index.html Theorem 12 and succeeding remarks.

Title Brauer group BrauerGroup 2013-03-22 11:49:31 2013-03-22 11:49:31 djao (24) djao (24) 13 djao (24) Definition msc 16K50 opposite algebra