The Classical Fields: Structural Features of the Real and Rational Numbers

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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.

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Editorial Reviews

From the Publisher
"Each of these fields comprise several closely interwoven algebraic and topological structures. Salzmann (mathematics, U. Tubingen) and his fellow authors Grundhofer (mathematics, U. Wurzburg, Hahl (mathematics, U. Stuttgart) and Lowen (mathematics, Technische U. Braunchwieg) explain the interaction and interdependencies of these fields. They begin with real numbers, describing real numbers as an ordered set and explaining the concept of real numbers as a field and as an ordered group and a topological group. They explain complex numbers and rational numbers, expanding on the latter as a field and describing rational numbers as a field, then work through the concept of completion, as chains, ordered groups, topological abelian groups, and topological rings and fields. They conclude by explaining the field of p-adic numbers, their squares, absolute values and valuations, the topologies of valuation type, local fields and locally compact fields. They provide exercises with hints and solutions." — Book News

"...self-contained...quite successful in fulfilling the state intentions. ...the detailed exposition of, for instance, group quotients and analytification could be of interest to more advanced students seeking through examples. It could well be listed as recommended secondary literature for a more traditional course in algebraic geometry." - Jon Eivind Vatne (Bergen), Mathematical Reviews

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Product Details

Meet the Author

Helmutt Salzmann is Full Professor of Mathematics at Mathematisches Institut, Universität Tübingen, Germany.

Theo Grundhöfer is Full Professor of Mathematics at Institut für Mathematik, Universität Würzburg, Germany.

Hermann Hähl is Full Professor of Mathematics at Institut für Geometrie und Topologie, Universität Stuttgart, Germany.

Rainer Löwen is Full Professor of Mathematics at Institut für Analysis und Algebra, Universität Braunschweig, Germany.

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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.

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