Classical Fields: Structural Features Of The Real And Ration

Description: The classical fields are the real, rational, complex and p-adic numbers. Each of these fields comprises several intimately interwoven algebraical and topological structures. This comprehensive volume analyzes the interaction and interdependencies of these different aspects. The real and rational numbers are examined additionally with respect to their orderings, and these fields are compared to their non-standard counterparts. Typical substructures and quotients, relevant automorphism groups and many counterexamples are described. Also discussed are completion procedures of chains and of ordered and topological groups, with applications to classical fields. The p-adic numbers are placed in the context of general topological fields: absolute values, valuations and the corresponding topologies are studied, and the classification of all locally compact fields and skew fields is presented. Exercises are provided with hints and solutions at the end of the book. An appendix reviews ordinals and cardinals, duality theory of locally compact Abelian groups and various constructions of fields. Review: '... certainly useful for readers with an interest in the interrelations of algebra and topology not based on abstract objects but on concrete ones, here reals and rationals.' European Mathematical Society Newsletter Table of Contents: Preface; 1. Real numbers; 2. Non-standard numbers; 3. Rational numbers; 4. Completion; 5. The p-adic numbers; 6. Appendix; Hints and solutions; Bibliography; Notation; Index. Author Biography: Helmutt Salzmann is Full Professor of Mathematics at Mathematisches Institut, Universitat Tubingen, Germany. Theo Grundhofer is Full Professor of Mathematics at Institut fur Mathematik, Universitat Wurzburg, Germany. Hermann Hahl is Full Professor of Mathematics at Institut fur Geometrie und Topologie, Universitat Stuttgart, Germany. Rainer Lowen is Full Professor of Mathematics at Institut fur Analysis und Algebra, Universitat Braunschweig, Germany.