I was wondering if anyone can suggest a reference which treats the Krull topology. Most of the books I have found don't go into any kind of detail.


It is my understanding that the Krull topology arises primarily as a way of obtaining a correspondence theorem for infinite Galois extensions. Are there other instances where this topology arises naturally? 


Any pointers to books or papers will be warmly received.



References and applications involving the Krull Toplogy
I think what you really want to read about, to see things in a bigger context, are profinite groups. A profinite group is a certain class of compact groups which includes Galois groups with their Krull topology. See Chapter 1 of Serre's "Galois Cohomology" and Chapter 6 of Karpilovsky's "Topics in Field Theory" for a discussion of profinite groups aimed at later use in the setting of infinite Galois theory. Also look at Lenstra's treatment of profinite groups at <http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf>.


There are whole books on the topic of profinite groups, e.g., by Wilson and by Ribes and Zalesskii. Among profinite groups, the pro-p groups play the basic role that p-groups do among finite groups, and a book-length treatment on an important class of pro-p groups is the book "Analytic pro-p Groups" by Dixon, du Sautoy, Mann, and Segal. Reading a whole book on such topics might be more than you're ready to swallow at this point. 


Concerning your question of places other than Galois theory where the Krull topology naturally arises, a better question to ask is where else besides Galois theory the concept of a profinite group is used (because the label "Krull topology" is limited to automorphism groups of field extensions). There are applications of profinite groups in both number theory and group theory. Do a search on the Golod-Shafarevich theorem and its applications. 


Returning to Galois groups and their Krull topology, to really understand what's going on with that topology, you ought to study closely an example where the topology can also be experienced in a different way. The simplest important example for that is $p$-power cyclotomic extensions of $\mathbf Q$: ${\rm Gal}({\mathbf Q}(\mu\_{p^\infty})/{\mathbf Q})$ is isomorphic in a really concrete way to 
the group ${\mathbf Z}^\times\_p$ of units in the $p$-adic integers, and in particular the Krull topology on the Galois side matches the $p$-adic topology on the $p$-adic side: two automorphisms are close when the $p$-adic units they correspond to are $p$-adically close. (This isomorphism generalizes the natural isomorphism from ${\rm Gal}(\mathbf Q(\mu\_{p^n}))/\mathbf Q)$ to $({\mathbf Z}/p^n{\mathbf Z})^\times$.)  

Try to see concretely how the action of ${\mathbf Z}\_p^\times$ on $p$-power roots of unity works and why the subgroup $1 + p^n{\mathbf Z}\_p$ in $\mathbf Z\_p^\times$ corresponds to the intermediate field ${\mathbf Q}(\mu\_{p^n})$. I don't think you can really get a feel for infinite Galois theory without understanding this basic example, so in particular if you don't know what the $p$-adic integers are then stop and learn about them, and then go back to your Krull topology studies. 


