PREFACE

Summary

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.