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Classical Fields Structural Features of the Real and Rational Numbers

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ISBN-10: 0521865166

ISBN-13: 9780521865166

Edition: 2007

Authors: H. Salzmann, T. Grundh�fer, H. H�hl, R. L�wen

List price: $185.00
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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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Book details

List price: $185.00
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 8/23/2007
Binding: Hardcover
Pages: 418
Size: 6.00" wide x 9.00" long x 1.00" tall
Weight: 1.694
Language: English

Helmutt Salzmann is Full Professor of Mathematics at Mathematisches Institut, Universit�t T�bingen, Germany.

Theo Grundh�fer is Full Professor of Mathematics at Institut f�r Mathematik, Universitat W�rzburg, Germany.

Hermann Hahl is Full Professor of Mathematics at Institut f�r Geometrie und Topologie, Universitat Stuttgart, Germany.

Rainer L�wen is Full Professor of Mathematics at Institut f�r Analysis und Algebra, Universitat Braunschweig, Germany.

Real numbers
The additive group of real numbers
The multiplication of real numbers, with a digression on fields
The real numbers as an ordered set
Continued fractions
The real numbers as a topological space
Characterizing the real line, the arc, and the circle
Independence of characteristic properties
Subspaces and continuous images of the real line
Homeomorphisms of the real line
Weird topologies on the real line
The real numbers as a field
The real numbers as an ordered group
The real numbers as a topological group
Subgroups and quotients
A counter-example
Automorphisms and endomorphisms
Groups having an endomorphism field
Multiplication and topology of the real numbers
The real numbers as a measure space
The real numbers as an ordered field
Formally real and real closed fields
The real numbers as a topological field
The complex numbers
Non-standard numbers
Non-standard rationals
A construction of the real numbers
Non-standard reals
Ordering and topology
Continuity and convergence
Topology of the real numbers in non-standard terms
Planes and fields
Rational numbers
The additive group of the rational numbers
The multiplication of the rational numbers
Ordering and topology of the rational numbers
The rational numbers as a field
Ordered groups of rational numbers
Addition and topologies of the rational numbers
Multiplication and topologies of the rational numbers
Completion of chains
Completion of ordered groups and fields
Completion of topological abelian groups
Completion of topological rings and fields
The p-adic numbers
The field of p-adic numbers
The additive group of p-adic numbers
The multiplicative group of p-adic numbers
Squai-es of p-adic numbers and quadratic forms
Absolute values
Topologies of valuation type
Local fields and locally compact fields
Ordinals and cardinals
Topological groups
Locally compact abelian groups and Pontryagin duality
Hints and solutions