# quaternion algebra

A 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.

Title quaternion algebra QuaternionAlgebra 2013-03-22 12:37:54 2013-03-22 12:37:54 djao (24) djao (24) 5 djao (24) Definition msc 11R52 msc 16K20