Ramification of a quaternion algebra at a place

Let $B$ be a quaternion algebra over a number field $K$ , and let $v$ be a place of $K$ , with completion $K_v$ . The algebra is ramified at $v$ if

B_v=B\otimes_K K_v

is a division algebra. It is split at $v$ if $B_v\cong M_2(K_v)$ . These are the only two possibilities, and $\operatorname{Ram}(B)$ denotes the set of ramified places.

Finite and infinite places

At a real place, ramification means $B_v\cong\mathbb H$ , as in real-place ramification. A complex place is always split. At a finite place, the definition concerns the quaternion algebra over the corresponding nonarchimedean local field.

For a number field, $\operatorname{Ram}(B)$ is finite, has even cardinality, and determines $B$ up to $K$ -algebra isomorphism.

Rational Hamilton example

For $B=(-1,-1)_{\mathbb Q}$ ,

\operatorname{Ram}(B)=\{2,\infty\}.

At odd primes the parameters are units and the local Hilbert symbol is $+1$ . At $2$ and $\infty$ it is $-1$ . The algebra discriminant records the finite prime $2$ , whereas the full ramification set also records $\infty$ .

References

John Voight, Quaternion Algebras, author's text, Definition 14.5.1, Lemma 14.5.3, and Main Theorem 14.6.1; §§12.4.12–12.4.14 for the local calculation.