Over number fields, finite dimensional central division algebras are always cyclic algebras. So the construction of cyclic algebras is a nice recipe to create algebras, which exhausts all finite dimensional central division algebras over number fields.
Is there a similar recipe for finite dimensional central division algebras over functions fields (by which I mean finite extensions of $\mathbb{F}_q(T)$)?
If not, and this is a somewhat different question, what are the some general families of such algebras, and are they known not to exhaust everything? Further, what if we restrict only to $4$-dimensional such algebras?
All central simple algebras over a global field are cyclic algebras. In Milne's class field theory notes he treats the case of number fields in section 2 of chapter VIII as a consequence of a corollary on the existence of a cyclic extension of a number field with finitely many specified local degrees. That same existence result on cyclic extensions is handled for all global fields as Theorem 5 in Chapter 10 (Grunwald's Theorem) of the Artin-Tate notes on class field theory.