Field arithmetic. (English) Zbl 0625.12001

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 11. A series of Modern Surveys in Mathematics. Berlin etc.: Springer-Verlag. xvii, 458 p. DM 198.00 (1986).
In recent years starting from work of James Ax and Abraham Robinson methods from “logic” emerged into other areas like field theory, number theory and arithmetic. Some of these developments gave valuable new insights to old questions of number theory and arithmetic, some of them were leading to new concepts. Most of the results so far were scattered through the literature, no coherent treatment of the subject was available. Obviously the authors’ intention is a detailed and self contained presentation of part of these developments.
Main focus and the central topic of the book is the theory of elementary statements in the sense of first order logic of certain classes of fields, a program, which partly developed by the authors themselves, provides interesting links between the two mathematical disciplines logic and arithmetic. As the authors claim themselves, the book should serve as a bridge between the two fields. Hence much stress is laid on thorough foundations. It is not an easy task to give comprehensive introductions to both subjects in one book. The result however should not only attract the specialist from one of these fields. It should serve as well as textbook for graduate courses or as an excellent survey suitable for personal studies. Almost every chapter has exercises and interesting notes on related literature at the end. The book also includes an appendix on open research problems.
As already mentioned it contains many chapters developing background concepts especially from elementary algebraic geometry with special emphasis on effective methods. The reader will find chapters on infinite Galois theory and profinite groups, algebraic function fields in one variable and plane curves. Two chapters are devoted to give complete and elementary proofs for the Chebotarev density theorem and the Riemann hypothesis for function fields. On the other hand we find chapters on ultraproducts, decision procedures, elementary theory of algebraically closed fields, undecidability, nonstandard model theory and a nonstandard proof of Hilbert’s irreducibility theorem. This foundational material more or less covers the first 15 chapters of the book.
All this is brought together in the study of pseudo-algebraically closed fields (PAC fields). PAC fields are fields \(K\) with the property, that every absolutely irreducible variety over \(K\) has a rational point. Such fields first arose from work of James Ax on the elementary theory of finite fields. They have connections with number theory because almost everywhere statements of first order logic on primes for algebraic number fields are related to the theory of a special class of PAC fields, those arising as ultraproducts of finite fields. To a certain degree this connection extends to statements which are true for sets of primes with given Dirichlet densities. Here we meet another important class of PAC fields which arise as fixed fields under finitely many generic (in the sense of Haar measure) automorphisms of the separable closure of a number field or more generally a countable Hilbertian field. This and related problems are discussed in chapters 16, 18, 19 and in refined form in chapter 26 of the book. In these chapters PAC fields often play the role of model theoretic completions of the underlying arithmetic of the given number field.
There are further chapters studying PAC fields on their own right. Here arithmetic seems to be almost lost. The main structure under consideration is the absolute Galois group of the PAC field, an interesting group which is known to be projective. A detailed discussion on decidability and undecidability results arising from that can be found in chapters 20, 21 and 22. The final chapters of the book are devoted to PAC fields, whose absolute Galois groups have the embedding property.


12-02 Research exposition (monographs, survey articles) pertaining to field theory
12L15 Nonstandard arithmetic and field theory
03C60 Model-theoretic algebra
12L12 Model theory of fields
12E30 Field arithmetic
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
03C10 Quantifier elimination, model completeness, and related topics
03C20 Ultraproducts and related constructions
14G05 Rational points
12L05 Decidability and field theory
03H15 Nonstandard models of arithmetic