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AuthorNarkiewicz, Wล{130}adysล{130}aw. author
TitleElementary and Analytic Theory of Algebraic Numbers [electronic resource] / by Wล{130}adysล{130}aw Narkiewicz
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2004
Edition Third Edition
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Descript XI, 712 p. online resource


The aim of this book is to present an exposition of the theory of algeยญ 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 numeriยญ 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 clasยญ sical theory of algebraic numbers is developed. Chapter 5 contains the funยญ 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 Shafareยญ vich's proof of the Kronecker-Weber theorem, and also the main properties of adeles and ideles


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. Bakerโ{128}{153}s Method -- Problems -- References -- Author Index -- List of Symbols

Mathematics Algebra Number theory Mathematics Number Theory Algebra


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