Classical Fields Structural Features of the Real and Rational Numbers

Name: Classical Fields Structural Features of the Real and Rational Numbers
Price: 51.38 USD
Availability: InStock
ISBN: 9780521865166

ISBN-10: 0521865166

ISBN-13: 9780521865166

Edition: 2007

Authors: H. Salzmann, T. Grundhï¿½fer, H. Hï¿½hl, R. Lï¿½wen

List price: $151.00

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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…

Book details

List price: $151.00
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 8/23/2007
Binding: Hardcover
Pages: 418
Size: 6.46" wide x 9.53" long x 0.98" tall
Weight: 1.650
Language: English

Helmutt Salzmann is Full Professor of Mathematics at Mathematisches Institut, Universitï¿½t Tï¿½bingen, Germany.

Theo Grundhï¿½fer is Full Professor of Mathematics at Institut fï¿½r Mathematik, Universitat Wï¿½rzburg, Germany.

Hermann Hahl is Full Professor of Mathematics at Institut fï¿½r Geometrie und Topologie, Universitat Stuttgart, Germany.

Rainer Lï¿½wen is Full Professor of Mathematics at Institut fï¿½r Analysis und Algebra, Universitat Braunschweig, Germany.



Preface


Notation



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


Characterizations


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



Ultraproducts



Non-standard rationals



A construction of the real numbers



Non-standard reals


Ordering and topology


[eta]1-fields



Continuity and convergence



Topology of the real numbers in non-standard terms



Differentiation



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



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



Valuations



Topologies of valuation type



Local fields and locally compact fields



Appendix



Ordinals and cardinals



Topological groups



Locally compact abelian groups and Pontryagin duality



Fields


Hints and solutions


References


Index