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

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

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

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

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Overview

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.

Product Details

ISBN-13: 9780521865166
Publisher: Cambridge University Press
Publication date: 08/23/2007
Series: Encyclopedia of Mathematics and its Applications , #112
Pages: 418
Product dimensions: 6.46(w) x 9.53(h) x 0.98(d)

About 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.

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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