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 $n^2$?

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.
https://www.jmilne.org/math/CourseNotes/CFT310.pdf
