# Quaternion algebras over a non-Archimedean local field $K$, up to isomorphism

I want to know the number of non-isomorphic quaternion algebras over a non-Archimedean local field $$K$$. What is the number of non-isomorphic central simple algebras of dimension $$n^2$$ over a non-Archimedean local field $$K$$?

I know the Brauer group of $$K$$ is isomorphic to $$\dfrac{\mathbb{Q}}{\mathbb{Z}}$$. I know the structure of the group $$\dfrac{\mathbb{Q}}{\mathbb{Z}}$$ very well, and it has only one element of order $$2$$.

Let $$n \in \mathbb{N}$$ be arbitrary. Is there any relation between the elements of order $$n$$ (or elements of order dividing $$n$$) in the group $$\dfrac{\mathbb{Q}}{\mathbb{Z}}$$, and the central simple algebras of dimension $$n^2$$?

• I think the order (squared) of a CSA in the Brauer group divides the dimension (unless I have things backwards) but they aren’t always equal. Perhaps look in Serre first since if there is a simple relationship it would very likely be there
– user208649
Oct 2, 2020 at 22:32

The elements of order $$n$$ in $$\frac{\mathbb{Q}}{\mathbb{Z}}$$ correspond bijectively to the isomorphism classes of central simple algebras over $$K$$ of dimension $$n^2$$. In particular there is a unique isomorphism class of (non-split) quaternion algebra. See Remark 4.4 on p. 110 here for an explicit construction.
• So can we conclude that there are exactly two equivalence classes of non-split algebras of dimension $9$? Am I right? Oct 3, 2020 at 7:40