A major, and relatively recent, topic in the arithmetic of fields has been the gradual introduction of the language and techniques of mathematical logic: ultraproducts, model theory, non-standard structures, decision problems and effective methods. These have shed new light on important questions and areas of arithmetic algebraic geometry and at the same time have created new concepts leading to a body of theory that provides a bridge between these parts of mathematics and logic. Before the first edition of this book (Springer, 1986) there was no monograph devoted to these aspects of field theory, and the book collected and organized these new results in one place in a coherent and systematic way.
The first six chapters of the book give an introduction to the arithmetic of fields that is fairly standard, covering infinite Galois theory, profinite groups, extensions of valued fields, algebraic function fields (including the Riemann-Roch theorem, zeta functions and the Riemann hypothesis, proved using Bombieri’s approach), Dirichlet density and the Chebotarev density theorem, and an introduction to affine and projective curves providing a geometric interpretation for results formulated in the language of function fields. But, even in this well-traveled road, they add something new, e.g., two proofs of the realization of all profinite groups as Galois groups of some Galois extension of some field (Leptin's Theorem)..
The next two chapters introduce the reader to some parts of mathematical logic that will be relevant later on: First order predicate calculus, structures and models, ultrafilters and ultraproducts, deduction theory, recursion, decidability and Gödel’s completeness theorem. Chapter 14 gives a short introduction to non-standard structures.
The first indication that the theory just developed can say something new comes in chapter nine, which provides an elementary (in the sense of first-order logic) theory of algebraically closed fields. In particular the authors prove that the theory of algebraically closed fields of a given characteristic is complete, and as an application obtain the weak version of Hilbert’s Nullstellensatz: If the ideal generated by a set of polynomials in n variables over a given field is not the whole ring, then the polynomials have a common zero in an algebraic closure of the given field. One important variation of this theorem goes as follows: If K is a non-principal ultraproduct of finite fields and if f is a non-constant absolutely irreducible polynomial in K[X,Y], then the curve defined by f has a K-rational point. Moreover, K has the stronger property that each non-empty algebraic variety defined over K has a K-rational point.
Thus, after recalling in chapter ten some basic elements of Algebraic Geometry (affine and projective varieties over given field) we come to one of the main concepts treated in this book, namely pseudo algebraically closed (PAC) fields, i.e., fields over which every non-empty algebraic variety has a rational point. It turns out that the PAC property can be verified on special varieties, namely open subsets of hypersurfaces (the primitive element theorem), open subsets of plane curves (intersecting with hypersurfaces) or any affine hypersurface. It also turns out that PAC fields are infinite and cannot not be ordered. After introducing the class of PAC fields, one of the goals is to understand the absolute Galois group of the given field, and one of the main results is the characterization of these Galois groups amongst the profinite groups: They are the projective profinite groups and as a consequence the Brauer group of a PAC field is trivial.
The second class of fields studied in this book is the class of Hilbertian fields, i.e., those fields that satisfy the conclusion of Hilbert’s irreducibility theorem. The main application is for the inverse Galois problem: Given a finite group G and a Hilbertian field K, first realize G as a Galois group over a transcendental extension of K and then use the Hilbertian property to realize G over the given field K. A major result is then a characterization of which PAC fields are Hilbertian, and a characterization of those PAC fields whose absolute Galois groups have the embedding property.
It should be clear now that the main objective of this monograph is to study the interrelations amongst the three major concepts just recalled, and about two thirds of the book is dedicated to this goal.
This remarkable monograph has gone through three editions since first published in 1986. The first edition had 26 chapters and 458 pages, the second enlarged edition (Springer, 2005) had 32 chapters and 780 pages. The third edition is a minor revision of the second one, correcting some typographical errors, especially regarding the bibliography. The list of problems at the end of the book has been updated taking in consideration the ones that have already been solved or partially solved.
There could be different uses of this book: It could be used a text for graduate students entering the field, since the material is so well organized, even including exercises at the end of every chapter. The book could be used as a nice introduction to the algebraic theory of fields (chapters 1 to 6) or as an introduction to mathematical logic with selected applications to field theory (chapters 7, 8 and 14) including an elementary introduction to the language of algebraic geometry (chapter 10). However, the book is mainly a monograph surveying some major advances in the arithmetic of fields, with its stronger part the chapters 11 to 32.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org