Lectures on the Model Theory of Valued Fields
The subject originates in the 1950s with Abraham Robinson when he established the model completeness of the theory of algebraically closed valued fields. In the 1960s Ax & Kochen and, independently, Ershov, proved a remarkable theorem on henselian val
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1 Introduction The subject originates in the 1950s with Abraham Robinson when he established the model completeness of the theory of algebraically closed valued fields. In the 1960s Ax & Kochen and, independently, Ershov, proved a remarkable theorem on henselian valued fields, with applications to p-adic number theory. These results and their refinements and extensions remain important in more recent developments like motivic integration. The AKE-theorems—with AKE abbreviating Ax, Kochen, Ershov—were the starting point for further work by many others. These lectures (given in the fall semester of 2004 at the University of Illinois at Urbana-Champaign, and augmented on the occasion of my talks at the CIME meeting in July 2012) have the modest goal of giving a transparent treatment of the original AKE-theorems and of Robinson’s result, including the relevant background on local rings and valuation theory. We assume only very basic knowledge: from algebra some familiarity with rings and fields (including some Galois theory) and from model theory: compactness, saturation, back-and-forth, model-theoretic tests for quantifier elimination. We do go beyond the original AKE-results by also paying attention to definable sets, which are at the center of present applications. In some of the key arguments in Chaps. 4 and 5 we use the technique of pseudocauchy sequences. This old technique originates with Ostrowski and was also used by AKE. There are ways to avoid this and give more constructive proofs, but in recent extensions to suitable valued difference fields and valued differential fields, it was this approach, suitably elaborated, that was successful. It is also very much in the spirit of model theory.
L. van den Dries () Department of Mathematics, University of Illinois, Urbana, IL, USA e-mail: [email protected] L. van den Dries et al., Model Theory in Algebra, Analysis and Arithmetic, Lecture Notes in Mathematics 2111, DOI 10.1007/978-3-642-54936-6__4, © Springer-Verlag Berlin Heidelberg 2014
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With just the techniques from the present notes we can do much more. With sharper model theoretic tools one can even go far beyond the scope of these notes; see for example [30] and [28]. As a partial compensation for our limited material, we give at the end more historical background and a list of references with comments. This includes pointers to developments closely related to the more classical material presented here. It should also be clear from the other lectures at this meeting that valuation theory currently plays a significant role in a variety of model-theoretic topics. I thank the referee and the editors for their careful reading of the manuscript. Implementing their suggestions improved the final product considerably. Conventions. Throughout, m; n range over N D f0; 1; 2; : : : g, the set of natural numbers. Unless specified otherwise, “ring” means “commutative ring with 1” and given a ring R we let U.R/ WD fx 2 R W xy D 1 for some y 2 Rg be its multiplicative group of units, and for
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