The Classical Fields

Structural Features of the Real and Rational Numbers

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Hardcover | 23 August 2007

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418 Pages

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22.86 x 15.24 x 2.54

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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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"Each of these fields comprise several closely interwoven algebraic and topological structures. Salzmann (mathematics, U. Tubingen) and his fellow authors Grundhofer (mathematics, U. Wurzburg>, Hahl (mathematics, U. Stuttgart) and Lowen (mathematics, Technische U. Braunchwieg) explain the interaction and interdependencies of these fields. They begin with real numbers, describing real numbers as an ordered set and explaining the concept of real numbers as a field and as an ordered group and a topological group. They explain complex numbers and rational numbers, expanding on the latter as a field and describing rational numbers as a field, then work through the concept of completion, as chains, ordered groups, topological abelian groups, and topological rings and fields. They conclude by explaining the field of p-adic numbers, their squares, absolute values and valuations, the topologies of valuation type, local fields and locally compact fields. They provide exercises with hints and solutions." -- Book News

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Preface	p. ix
Notation	p. xiii
Real numbers	p. 1
The additive group of real numbers	p. 2
The multiplication of real numbers, with a digression on fields	p. 14
The real numbers as an ordered set	p. 22
Continued fractions	p. 28
The real numbers as a topological space	p. 32
Characterizing the real line, the arc, and the circle	p. 34
Independence of characteristic properties	p. 44
Subspaces and continuous images of the real line	p. 51
Homeomorphisms of the real line	p. 57
Weird topologies on the real line	p. 64
The real numbers as a field	p. 70
The real numbers as an ordered group	p. 75
The real numbers as a topological group	p. 81
Subgroups and quotients	p. 83
Characterizations	p. 86
A counter-example	p. 93
Automorphisms and endomorphisms	p. 95
Groups having an endomorphism field	p. 96
Multiplication and topology of the real numbers	p. 100
The real numbers as a measure space	p. 104
The real numbers as an ordered field	p. 112
Formally real and real closed fields	p. 122
The real numbers as a topological field	p. 135
The complex numbers	p. 140
Non-standard numbers	p. 154
Ultraproducts	p. 154
Non-standard rationals	p. 158
A construction of the real numbers	p. 159
Non-standard reals	p. 162
Ordering and topology	p. 164
[eta]1-fields	p. 166
Continuity and convergence	p. 170
Topology of the real numbers in non-standard terms	p. 173
Differentiation	p. 175
Planes and fields	p. 177
Rational numbers	p. 179
The additive group of the rational numbers	p. 179
The multiplication of the rational numbers	p. 185
Ordering and topology of the rational numbers	p. 193
The rational numbers as a field	p. 207
Ordered groups of rational numbers	p. 216
Addition and topologies of the rational numbers	p. 221
Multiplication and topologies of the rational numbers	p. 228
Completion	p. 235
Completion of chains	p. 236
Completion of ordered groups and fields	p. 239
Completion of topological abelian groups	p. 248
Completion of topological rings and fields	p. 264
The p-adic numbers	p. 278
The field of p-adic numbers	p. 279
The additive group of p-adic numbers	p. 285
The multiplicative group of p-adic numbers	p. 292
Squai-es of p-adic numbers and quadratic forms	p. 295
Absolute values	p. 300
Valuations	p. 306
Topologies of valuation type	p. 316
Local fields and locally compact fields	p. 322
Appendix	p. 335
Ordinals and cardinals	p. 335
Topological groups	p. 340
Locally compact abelian groups and Pontryagin duality	p. 344
Fields	p. 350
Hints and solutions	p. 360
References	p. 383
Index	p. 399
Table of Contents provided by Ingram. All Rights Reserved.

The Classical Fields

Structural Features of the Real and Rational Numbers

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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