I should start by saying that I have not studied field theory in depth, so if this question is totally off base, I apologize. Something I noticed as I studied group theory is many concepts that were very difficult to define directly had simple and elegant categorical definitions. For example, the direct definition of the free group is rather long and arduous, whereas the categorical definition, i.e. any function $S\to G$, where $G$ is a group factors through a homomorphism from the free group generated by $S$ to $G$, is quite simple. However, for the most part, it seems to me that categorical methods are most easily used on infinite groups, and in particular, infinite abelian groups. Despite this limitation, categorical methods seemed so natural that I couldn't help but wonder if they can be applied to field theory with similar results. So my question is: (1) is it beneficial to study infinite field theory in the generality that category theory necessitates, and (2) are there any good books that use this approach.



Infinite Field Theory and Category Theory
I think Mike Skirvin's comment above should be expanded into an answer.
There are no homomorphisms at all between fields of different characteristic. Hence one has to look at the category of fields of a fixed characteristic $p$.  

An elementary fact about fields is that they have no nontrivial ideals.  

It follows that all homomorphisms between fields are 1-1. 
This implies that there are no free fields of any characteristic $p$
(except for the free field of char $p$ over the empty set of generators). 


Finally, as Mike Skirvin pointed out in his comment, there are in general
no products of fields, even of a fixed characteristic.
I think this sufficiently explains why categorial constructions are not very
useful in field theory. 


