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  3. The Classical Fields
  • Cited by 14
  • 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
Publisher:
Cambridge University Press
Online publication date:
May 2010
Print publication year:
2007
Online ISBN:
9780511721502
DOI:
https://doi.org/10.1017/CBO9780511721502
Subjects:
Mathematics (general), Mathematics, Geometry and Topology, Algebra
Series:
Encyclopedia of Mathematics and its Applications (112)
Information

Book 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.

Reviews

'… 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.'

Source: European Mathematical Society Newsletter

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