Valuations are among the fundamental structures of number theory and of algebraic geometry. This was recognized early by model theorists, with gratifying results: Robinson's description of algebraically closed valued fields as the model completion of the theory of valued fields; the Ax-Kochen, Ershov study of Henselian fields of large residue characteristic with the application to Artin's conjecture work of Denef and others on integration; and work of Macintyre, Delon, Prestel, Roquette, Kuhlmann, and others on p-adic fields and positive characteristic. The model theory of valued fields is thus one of the most established and deepest areas of the subject.

However, precisely because of the complexity of valued fields, much of the work centers on quantifier elimination and basic properties of formulas. Few tools are available for a more structural model-theoretic analysis. This contrasts with the situation for the classical model complete theories, of algebraically closed and real closed fields, where stability theory and o-minimality make possible a study of the category of definable sets. Consider for instance the statement that fields interpretable over ℂ are finite or algebraically closed. Quantifier elimination by itself is of little use in proving this statement. One uses instead the notion of ω-stability; it is preserved under interpretation, implies a chain condition on definable subgroups, and, by a theorem of Macintyre, ω-stable fields are algebraically closed. With more analysis, using notions such as generic types, one can show that indeed every interpretable field is finite or definably isomorphic to ℂ itself.

We give here a brief preview of stability theory, as it underpins stable domination. We also introduce some of the model-theoretic notation used later. Familiarity with the basic notions of logic (languages, formulas, structures, theories, types, compactness) is assumed, but we explain the model theoretic notions beginning with saturation, algebraic closure, imaginaries. We have in mind a reader who is familiar with o-minimality or some model theory of valued fields, but has not worked with notions from stability. Sources include Shelah's Classification Theory as well as books by Baldwin, Buechler, Pillay and Poizat. There is also a broader introduction by Hart intended partly for non-model theorists, and an introduction to stability theory intended for a wider audience in. Most of the stability theoretic results below should be attributed to Shelah. Our treatment will mostly follow Pillay.

Stability theory is a large body of abstract model theory developed in the 1970s and 1980s by Shelah and others, but having its roots in Morley's 1965 Categoricity Theorem: if a complete theory in a countable language is categorical in some uncountable power, then it is categorical in all uncountable powers. Shelah formulated a radical generalization of Morley's theorem, weakening the categoricity assumption from one isomorphism type to any number less than the set-theoretic maximum. The conclusion is that all models of the theory, in any power, are classifiable by a small tree of numerical dimensions.

As developed in, stability theory is based on the notion of an invariant type, more specifically a definable type, and the closely related theory of independence of substructures. We will review the definitions in Chapter 2 below; suffice it to recall here that an (absolutely) invariant type gives a recipe yielding, for any substructure A of any model of T, a type p│A, in a way that respects elementary maps between substructures; in general one relativizes to a set C of parameters, and considers only A containing C. Stability arose in response to questions in pure model theory, but has also provided effective tools for the analysis of algebraic and geometric structures. The theories of algebraically and differentially closed fields are stable, and the stability-theoretic analysis of types in these theories provides considerable information about algebraic and differential-algebraic varieties. The model companion of the theory of fields with an automorphism is not quite stable, but satisfies the related hypothesis of simplicity; in an adapted form, the theory of independence remains valid and has served well in applications to difference fields and definable sets over them. On the other hand, such tools have played a rather limited role, so far, in o-minimality and its applications to real geometry.

Where do valued fields lie? Classically, local fields are viewed as closely analogous to the real numbers. We take a “geometric” point of view however, in the sense of Weil, and adopt the model completion as the setting for our study.