Infinite Field Theory and Category Theory 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.
 A: Category theory can be useful for certain aspects of (infinite) field theory, but really requires that you do not restrict just to fields. I am meaning modern approaches to Galois theory. That is not really field theory as such but intersects very nicely with that area. To see how category theory interacts with field theory look at the book by Borceux and Janelidze:
 Galois theories , volume 72 of Cambridge Studies in Advanced Mathematics , Cambridge University Press.
That goes from a fairly classical viewpoint to a categorical one, but the classical one is, of course, looked at from a categorical viewpoint. 
A: 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.  
A: The category of fields is a full subcategory of the category of rings. The latter has nice categorical properties, but the subcategory does not inherit these properties. Of course, you may apply every theorem of category theory to the category of fields, but the problem is, that homomorphisms of fields are rather rigid. If $F$ is a field, it is very rare that there is a nice description for the Hom-functor $Hom(F,-)$. For example, if $F=\mathbb{Q}(t)$ is a function field, then $Hom(F,L)$ is empty if $char(L)>0$, and otherwise can be identified with the elements of $L$, which are transcendental over $\mathbb{Q}$. Anyway, in the theory of algebraic extensions, it is good to know that $Hom_K(K(\alpha),L)$ consists of the roots of $\alpha$ in $L$ of the minimal polynomial of $\alpha$ over $K$. But this already takes place in the category of rings.
Let $K$ be a field. It is a useful fact that the category of field extensions of $K$ is directed in the sense that for every pair of extensions $L/K, L'/K$ there is an extension $M/K$ together with $K$-homomorphisms $L \to M, L' \to M$. Namely, $L \otimes_K L'$ is a nontrivial ring, so we may mod out some maximal ideal to get such a field. The same works for infinitely many extensions and can be used to show the existence of an algebraic closure of $K$. Note that the algebraic closure does not have a honest universal property, but it's construction reminds of other universal objects. The fact that the extensions over $K$ is directed also implies that for schemes $X,Y$ over $S$, the natural map $|X \times_S Y| \to |X| \times_{|S|} |Y|$ is surjective (where $|X|$ is the underlying topological space of $X$).
I suspect that there are applications of category theory to the category of fields, as the others already have pointed out. However, sometimes this category can be useful: Let $X$ be a scheme. Then the colimit of the $X(K)$, where $K$ runs through the category of fields, is just the underlying set of $X$. This is useful when you try to recover a scheme just by a given universal property.
