Generalization
Given two division rings E and F with F contained in E and the multiplication and addition of F being the restriction of the operations in E, we can consider E as a vector space over F in two ways: having the scalars act on the left, giving a dimension l, and having them act on the right, giving a dimension r. The two dimensions need not agree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above applies to left-acting scalars without change.
Read more about this topic: Degree Of A Field Extension