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The Classical Fields

The Classical Fields
Structural Features of the Real and Rational Numbers

Part of Encyclopedia of Mathematics and its Applications

  • Authors:
  • H. Salzmann, Eberhard-Karls-Universität Tübingen, Germany
  • T. Grundhöfer, Bayerische-Julius-Maximilians-Universität Würzburg, Germany
  • H. Hähl, Universität Stuttgart
  • R. Löwen, Technische Universität Carolo Wilhelmina zu Braunschweig, Germany
  • Date Published: August 2007
  • availability: Available
  • format: Hardback
  • isbn: 9780521865166


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About the Authors
  • 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.

    • First book to comprehensively discuss the abstract structural properties of the classical number systems of mathematics
    • Discusses in detail the interrelations between real, rational, complex and p-adic numbers
    • Contains over 200 exercises, hints and solutions
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    Reviews & endorsements

    '… 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

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    Product details

    • Date Published: August 2007
    • format: Hardback
    • isbn: 9780521865166
    • length: 418 pages
    • dimensions: 242 x 164 x 25 mm
    • weight: 0.754kg
    • contains: 205 exercises
    • availability: Available
  • Table of Contents

    1. Real numbers
    2. Non-standard numbers
    3. Rational numbers
    4. Completion
    5. The p-adic numbers
    6. Appendix
    Hints and solutions

  • Authors

    H. Salzmann, Eberhard-Karls-Universität Tübingen, Germany
    Helmutt Salzmann is Full Professor of Mathematics at Mathematisches Institut, Universität Tübingen, Germany.

    T. Grundhöfer, Bayerische-Julius-Maximilians-Universität Würzburg, Germany
    Theo Grundhöfer is Full Professor of Mathematics at Institut für Mathematik, Universität Würzburg, Germany.

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

    R. Löwen, Technische Universität Carolo Wilhelmina zu Braunschweig, Germany
    Rainer Löwen is Full Professor of Mathematics at Institut für Analysis und Algebra, Universität Braunschweig, Germany.

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