nt.number theory - References and applications involving the Krull Toplogy

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)