Algebraic view

Let $K$ be a field. The Brauer group $\operatorname{Br}(K)$ 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 $A$ .

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.html Theorem 12 and succeeding remarks.