Definition:Quaternion/Algebra over Field

Definition

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.