Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $\mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $\mathbb{Q}$-algebra embedding $K \hookrightarrow A$?

This is equivalent to asking whether $A \otimes_{\mathbb{Q}} K \simeq M_n(K)$. Is there a local-to-global principle that can be used?

If, for example, $A \simeq M_n(D)$ where $D$ is a quaternion algebra over $\mathbb{Q}$, is there a simple criterion relating subfields of $K$ and splitting fields of $D$?

Any reference to a similar example where this is worked out would be greatly appreciated!