I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's concentrate on these:
Which number fields $K$ occur as subfields of a finite-dimensional division algebra over $\mathbb{Q}$ with center $\mathbb{Q}$?
Which pairs of number fields $K, L$ occur as subfields of the same finite-dimensional division algebra over $\mathbb{Q}$ with center $\mathbb{Q}$?
There are some easy examples involving quaternions but I am curious how completely these kinds of questions are understood. Some preliminary Googling on my part was not successful.