Quaternion algebra

Summary

A quaternion algebra over a field $F$ is a central simple algebra over $F$ of dimension $4$. When the characteristic of $F$ is not $2$, it has a basis $1, i, j, k$ with $i^2 = a$, $j^2 = b$ and $ij = -ji = k$ for some nonzero $a, b \in F$. Each one is either a division algebra or isomorphic to the algebra of $2 \times 2$ matrices over $F$. The class of a quaternion algebra in the Brauer group has order at most $2$. Quaternion algebras are tied to quadratic forms and the Hilbert symbol.