Theory of ordered fields (Q2847544)

This is a self-contained and remarkably complete introductory text on the foundations of real algebra. Prerequisites consist of standard courses on linear algebra, groups, rings, fields and general topology.NEWLINENEWLINEThe chapter headings are as follows. 1. Formally real fields. 2. Quadratic forms. 3.~Real closed fields. 4. Ordered fields. 5. Space of orderings of formally real field. 6.~Valuation rings, valuations and places. 7. Valuation rings in formally real fields. 8.~Around Hilbert's 17 problem. 9.~Special classes of fields. 10.~Geometric properties of ordered fields. Appendix: Ordered Abelian groups; The field of real numbers. Each chapter ends with a list of well-balanced exercises, 180 altogether.NEWLINENEWLINEChapter 1 deals with the Artin-Schreier theory and develops the machinery of ordered fields, pre-orderings, signatures, fans, half-orderings, and extensions of orderings in field extensions. Chapter 2 gives an introduction to quadratic forms over fields including Pfister forms and trace forms, and Pfister's Local-Global Principle (without using the concept of the Witt ring). Chapter 3 introduces real closed fields and proves the existence and uniqueness of real closures. Characterization of real closed fields as subfields of algebraically closed fields of finite codimension is proved. Some elementary theorems of mathematical analysis (Weierstrass, Darboux, Rolle, Lagrange) are proved for rational functions over real closed field. In Chapter 4, dense, archimedean and continuous orderings are studied. Subfields of the field of real numbers and rational function fields as subfields of formal power series fields are the main source of examples. In Chapter 5 the boolean space of orderings of a field is studied. It is shown that the space is homeomorphic with the space of signatures of the field contained in a Cantor cube. Fields with Strong Approximation Property (SAP) are studied in detail. In Chapter 6 some basic notions and facts of valuation theory are presented. Convex subrings of ordered fields are described in algebraic terms. Approximation theorems are proved and extensions of valuation rings are discussed. In Chapter 7 the discussion of topics of Chapter 6 is continued in the context of formally real fields. The subheadings include formally real valuation rings (including the Baer-Krull theorem), Henselian valuation rings, real places and topological space of real places, localization of preorderings, half-orderings and valuation rings. In Chapter 8, a proof of Artin's theorem is given using Lang's approach. Two other results presented here are Positivstellensatz and Pfister's version of Hilbert's theorem on ternary forms of degree 4. Special classes of fields of Chapter 9 are Euclidean fields, Pythagorean fields, super-real and super-pythagorean fields. In each case various characterizations are given and some further questions are discussed (like going-up and going-down properties). The final Chapter 10 discusses links between main subject of the book and the linear algebra and geometry courses of the undergraduate curriculum. First, the fields \(K\) with the property that the spectral theorem for endomorphisms of bilinear spaces over \(K\) holds, are characterized. Second, Pasch fields are characterized in terms of half-orderings. Third, fields satisfying SAP are given a final touch. Fourth, characterizations are given for fields \(K\) with the property that Rolle's theorem holds for polynomials (or rational functions) over \(K\).NEWLINENEWLINEConcluding, the reader will be impressed by the wealth of material accommodated in the book and by the thoroughness of the discussion. Main results are well explained and motivated, and their significance is stressed when appropriate. This is a valuable textbook and it is going to be very comfortable to use it in an advanced undergraduate or graduate class, or in self-instruction.