quaternion algebra
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.
Title  quaternion algebra 

Canonical name  QuaternionAlgebra 
Date of creation  20130322 12:37:54 
Last modified on  20130322 12:37:54 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  5 
Author  djao (24) 
Entry type  Definition 
Classification  msc 11R52 
Classification  msc 16K20 