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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.

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2 Answers 2

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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.

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@K: Funny, I only observed just now that editing your response pushes it down to the bottom of the subqueue of equally voted responses. It happened to both of us... –  Pete L. Clark Mar 13 '10 at 0:46
    
@KConrad- Thanks for all the references and explanations. I picked up Karpilovsky's from the library and it looks useful. –  confusedmath Mar 13 '10 at 7:13
    
@Anyone- A quick search for profinite groups on google books turned up S.S. Shatz's "Profinite Groups, Arithmetic, and Geometry" which appears to discuss the Shafarevich-Golod theorem a fair bit. Are there any opinions about this book? –  confusedmath Mar 13 '10 at 7:15
    
Patrick: don't take Shatz as your first book on the Golod-Shafarevich (note the order of the names) theorem. Learn about profinite groups and infinite Galois theory elsewhere first and then, if tastes dictate, look at Shatz. –  KConrad Mar 13 '10 at 19:34

I recommend Chapter 8 of Jacobson's Basic Algebra II as a good general reference for the Krull topology and its applications in Galois theory. As is usual for BAI and BAII, if you read other books first you will get very excited at the depth of coverage of this topic.

A few remarks:

1) The Krull topology can actually be defined on $\operatorname{Aut}(E/F)$, for any field extension $E/F$. Indeed it is just the subspace topology it inherits from the compact open topology on the set of all maps from $E$ to $E$, where $E$ is given the discrete topology. (Strangely, this one topology -- the preferred function space topology in all my mathematical travels -- gets many names in this case. Jacobson calls it the finite topology. I have also heard it referred to as the "hull-kernel" topology -- ugh!.) If I am not mistaken, it is always totally disconnected and Hausdorff but need not be compact if $E/F$ is transcendental.

2) There have been some efforts (including by me!) to extend Galois theory to transcendental field extensions. The Krull topology comes up in the general case, e.g. in some papers of T. Soundararajan.

3) The Krull topology is also the topology associated to the Galois connection on $\operatorname{Aut}(E/F)$, hence it comes up in universal algebra, order theory, mathematical logic, etc. I don't know enough about these fields to point you to any particularly interesting application there, but someone else here certainly might.

Addendum: For instance, here is one of the papers I had in mind in 2) above:

Soundararajan, T. Galois theory for general extension fields. J. Reine Angew. Math. 241 1970 49--63.

The general aim of this paper is to investigate exhaustively general Galois correspondences on the level of fields. Specifically, the author considers correspondences when a topology is involved on the group of automorphisms.

Let $E$ be an extension of a field $K$ and $G_0$ the full group of $K$-automorphisms of $E$. We say $(E/K;G_0)$ is a Krull Galois system if there is a 1--1 Galois correspondence between all the intermediate fields of $E/K$ and all the ``Krull-closed'' sub-groups of $G_0$. It is classical that every separable algebraic normal extension allows a Krull Galois theory. The first theorem in this paper states the converse to the above. The author calls the triple $(E/K,G,\tau)$ a topological Galois system if there exists a 1--1 correspondence between all intermediate fields of $E/K$ and all $\tau$-closed sub-groups of $G$, where $\tau$ is some topology on $G$. Theorem 4 catalogues the conditions on the topology $\tau$ in order that $E$ may be separable algebraic normal over $K$, where $(E/K,G,\tau)$ is a topological Galois system. The next section deals with generalized topological Galois systems where there is a 1--1 Galois correspondence between all intermediate fields [all $\tau$-closed sub-groups of $G$] and some sub-groups of $G$ [some intermediate subfields of $E/K$]. The last section deals with a characterization of Krull topology and concludes with the following theorem: Let $(E/K,G,\tau_1)$ be a topological Galois system such that $(G,\tau_1)$ is a compact topological group. Then $E$ is separable algebraic normal over $K$, $G$ is the Galois group of $E/K$ and $\tau_1$ is the Krull topology.

This paper is lucidly presented and is highlighted by illustrative examples and useful remarks. (MathSciNet review by N. Sankaran)

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