A quaternion algebra over a field $K$ is a central simple algebra over $K$ which is four dimensional as a vector space over $K$ .

Examples:

• For any field $K$ , the ring $M_{2\times 2}(K)$ of $2\times 2$ matrices with entries in $K$ is a quaternion algebra over $K$ . If $K$ is algebraically closed, then all quaternion algebras over $K$ are isomorphic to $M_{2\times 2}(K)$ .
• For $K = \mathbb{R}$ , the well known algebra $\mathbb{H}$ of Hamiltonian quaternions is a quaternion algebra over $\mathbb{R}$ . The two algebras $\mathbb{H}$ and $M_{2 \times 2}(\mathbb{R})$ are the only quaternion algebras over $\mathbb{R}$ , up to isomorphism.
• When $K$ is a number field, there are infinitely many non-isomorphic quaternion algebras over $K$ . In fact, there is one such quaternion algebra for every even sized finite collection of finite primes or real primes of $K$ . The proof of this deep fact leads to many of the major results of class field theory.

One can show that every quaternion algebra over $K$ other than $M_{2\times 2}(K)$ is always a division ring.