Quaternion algebra over a field

Let $F$ be a field of characteristic different from two. A quaternion algebra over $F$ is an associative unital $F$ -algebra isomorphic to

(a,b)_F=F\oplus Fi\oplus Fj\oplus Fij,\qquad a,b\in F^\times,

whose multiplication is determined by $i^2=a$ , $j^2=b$ , and $ij=-ji$ . The four displayed elements form an $F$ -basis. The pair $(a,b)$ is a presentation, not an invariant unique to the algebra.

Examples

The real algebra $(-1,-1)_{\mathbb R}$ is the Hamilton quaternion algebra. Quaternion algebras need not be division algebras: the matrix algebra $M_2(F)$ is the split example.

Scope

This presentation is sufficient over number fields. Characteristic two uses different defining relations and is outside this knowl's convention.

References

F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Transactions of the AMS 349 (1997). Author-hosted paper, §4, presentation preceding Lemma 4.1.