Overview

The aim of this book is to present an exposition of the theory of alge­ braic numbers, excluding class-field theory and its consequences. There are many ways to develop this subject; the latest trend is to neglect the classical Dedekind theory of ideals in favour of local methods. However, for numeri­ cal computations, necessary for applications of algebraic numbers to other areas of number theory, the old approach seems more suitable, although its exposition is obviously longer. On the other hand the local approach is more powerful for analytical purposes, as demonstrated in Tate's thesis. Thus the author has tried to reconcile the two approaches, presenting a self-contained exposition of the classical standpoint in the first four chapters, and then turning to local methods. In the first chapter we present the necessary tools from the theory of Dedekind domains and valuation theory, including the structure of finitely generated modules over Dedekind domains. In Chapters 2, 3 and 4 the clas­ sical theory of algebraic numbers is developed. Chapter 5 contains the fun­ damental notions of the theory of p-adic fields, and Chapter 6 brings their applications to the study of algebraic number fields. We include here Shafare­ vich's proof of the Kronecker-Weber theorem, and also the main properties of adeles and ideles.

Full Product Details

Author:   Wladyslaw Narkiewicz
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of the original 3rd ed. 2004
Dimensions:   Width: 15.50cm , Height: 3.60cm , Length: 23.50cm
Weight:   1.098kg
ISBN:  

9783642060106

ISBN 10:   3642060102
Pages:   712
Publication Date:   04 December 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1. Dedekind Domains and Valuations.- 2. Algebraic Numbers and Integers.- 3. Units and Ideal Classes.- 4. Extensions.- 5. P-adic Fields.- 6. Applications of the Theory of P-adic Fields.- 7. Analytical Methods.- 8. Abelian Fields.- 9. Factorizations 9.1. 485Elementary Approach.- Appendix I. Locally Compact Abelian Groups.- Appendix II. Function Theory.- Appendix III. Baker’s Method.- Problems.- References.- Author Index.- List of Symbols.

Reviews

Aus den Rezensionen zur 3. Auflage: Das vorliegende Buch ist die dritte Auflage des Standardwerks uber analytische Aspekte der algebraischen Zahlentheorie. ! das Buch ist durchgehend in LaTeX gesetzt, was die Lekture angenehmer gestaltet. ! Durch die Aktualisierung wird dieses Buch sicher fur die nachsten Jahre ein wichtiges Referenzwerk in diesem Gebiet bleiben. (P. Grabner, in: IMN - Internationale Mathematische Nachrichten, 2006, Issue 202, S. 41 f.)

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