Every field extension can be factorized as a purely transcendental extension? followed by an algebraic extension. Indeed, by Zorn's lemma, we may construct a transcendence basis (i.e. maximal algebraically independent set) , and the purely transcendental part is the subfield generated by .
Unfortunately, this does not yield an orthogonal factorization system: given a field , we may form the field of rational functions over , which is a purely transcendental extension of , and we may form the algebraic closure , which is an algebraic extension of ; but we have the following commutative diagram,
where is the subfield of generated by , and is algebraic, yet there is no homomorphism making both evident triangles commute.