Discriminant of a quaternion algebra

Let $B$ be a quaternion algebra over a number field $K$ , and let $\mathcal O_K$ be its ring of integers. The discriminant of $B$ is the squarefree ideal

\operatorname{disc}(B)= \prod_{\substack{\mathfrak p\subset\mathcal O_K\\B\text{ ramified at }\mathfrak p}} \mathfrak p,

where the product runs over finite ramified places. The empty product is $\mathcal O_K$ .

Over the rational numbers

For $K=\mathbb Q$ , one usually writes the positive squarefree integer $D_B$ generating this ideal. The rational Hamilton algebra has $D_B=2$ , since its ramification set is $\{2,\infty\}$ . The split matrix algebra has $D_B=1$ .

What it records

The discriminant records finite ramification. It omits real places; the full ramification set is needed to classify quaternion algebras over general number fields.

This invariant belongs to the algebra. The reduced discriminant of an order depends on the chosen order, and agrees with the algebra discriminant for maximal orders.

References

John Voight, Quaternion Algebras, author's text, Definition 14.5.4, Remark 14.5.5, and §15.5.