\begin{definition}[Definition:Quaternion/Algebra over Field]
An algebra of quaternions can be defined over any field as follows:
Let $\mathbb K$ be a field, and $a$, $b \in \mathbb K$.
Define the '''quaternion algebra''' $\left\langle{ a,b }\right\rangle_\mathbb K$ to be the $\mathbb K$-vector space with basis $\{1, i, j, k\}$ subject to:
: $i^2 = a$
: $j^2 = b$
: $ij = k = -ji$
Formally this could be achieved as a multiplicative presentation of a suitable group, or as a linear subspace of a finite extension of $\mathbb K$.
Taking $\mathbb K = \R$ and $a = b = -1$ we see that this generalises  Hamilton's quaternions.
Category:Definitions/Quaternions
\end{definition}