Is there a specific reference for finite dimensional, associative, unital $\mathbb{Q}-$algebras that are division rings? Or also more in general, the type of questions I am trying to look into are:
How does the tensor product (over $\mathbb{Q}$) of two such rings look like? What is its center, depending on the centers of the two rings? Can I construct the compositum of two such rings and look at their intersection in there? How does it behave? What can be said about the Jacobson radical of such rings? How does the commutator of a subalgebra behave? What if the subalgebra is also a division ring? What is the commutator of the commutator? ...
I am particularly interested in knowing how the dimensions of all these $\mathbb{Q}-$algebras relate to one another. Any (not too general) reference or partial answer is appreciated.