Brauer group

1 Algebraic view

Let $K$ be a field. The Brauer group $\mathrm{Br}(K)$ 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 $\mathrm{Br}(K)$ is given by tensor product: for any two central simple algebras $A,B$ over $K$, theirproduct in $\mathrm{Br}(K)$ is the central simple algebra $A{\otimes }_{K}B$. The identity element in $\mathrm{Br}(K)$ is the class of $K$ itself, and the inverse of a central simple algebra $A$ isthe opposite algebra $A^{\mathrm{opp}}$ defined by reversingthe order of the multiplication operation of $A$.

2 Cohomological view

The Brauer group of $K$ is naturally isomorphic to the second Galois cohomology group$H^2(\mathrm{Gal}(K^{\mathrm{sep}}/K),{(K^{\mathrm{sep}})}^{\times })$.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.