I say that a skewfield (division ring) $D$ over $\mathbb{Q}$ is cyclotomic whenever $D$ admits a finite $\mathbb{Q}$-basis $\{\zeta_1,\dots,\zeta_n\}$ with each $\zeta_i$ a root of unity (i.e., $\zeta_i^{m_i}=1$ for some $m_i\geq1$). If we assume $D$ to be commutative, we know that $D$ is isomorphic to some $\mathbb{Q}(e^{2\pi i/m})$ and, in this case, $[D:\mathbb{Q}]=\dim_\mathbb{Q}(D)=\varphi(m)$. So my question is:
Q: What about the non-commutative case? Do we have a classification of the cyclotomic skewfields and do we have constraints or closed expressions for their degrees $[D:\mathbb{Q}]=\dim_\mathbb{Q}(D)$?
I am especially interested in cyclotomic skewfields that embed into the Hamilton quaternions $\mathbb{H}$ such as the cyclotomic fields or the rational quaternions $\left(\frac{-1,-1}{\mathbb{Q}}\right)$ of degree $4$. So, if there's a classification of cyclotomic sub-skewfields of $\mathbb{H}$ (up to conjugacy), I would be very interested to read it.
