Ramification of a quaternion algebra at a real place

Let $B$ be a quaternion algebra over a number field $K$ , and $\sigma:K\hookrightarrow\mathbb R$ a real embedding. Then $B$ is ramified at the real place $\sigma$ if

B\otimes_{K,\sigma}\mathbb R\cong\mathbb H,

the Hamilton division algebra. Otherwise this scalar extension is $M_2(\mathbb R)$ , and the algebra is split at that place. The tensor product uses $\sigma$ for the $K$ -algebra structure on $\mathbb R$ .

Sign test

For $B=(a,b)_K$ , ramification at $\sigma$ is equivalent to $\sigma(a)<0$ and $\sigma(b)<0$ . Rescaling the generators reduces that case to $(-1,-1)_{\mathbb R}$ . If either parameter is positive, its square root yields a split presentation.

Relation to all places

This is the real specialization of ramification at a place. A quaternion algebra over a totally real field that ramifies at every real place is totally definite.

Arithmetic significance

An imaginary quadratic field has no real embeddings, so “ramified at all real places” is vacuous there. This is why its split quaternion algebra is allowed in the arithmetic Kleinian construction.

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, Lemma 4.1.