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Ebook: Elementary and Analytic Theory of Algebraic Numbers

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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.




This book gives an exposition of the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. The following topics are treated: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. Each chapter ends with exercises and a short review of the relevant literature up to 2003. The bibliography  has over 3400 items.




This book gives an exposition of the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. The following topics are treated: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. Each chapter ends with exercises and a short review of the relevant literature up to 2003. The bibliography  has over 3400 items.


Content:
Front Matter....Pages I-XI
Dedekind Domains and Valuations....Pages 1-41
Algebraic Numbers and Integers....Pages 43-83
Units and Ideal Classes....Pages 85-133
Extensions....Pages 135-197
P-adic Fields....Pages 199-255
Applications of the Theory of P-adic Fields....Pages 257-312
Analytical Methods....Pages 313-407
Abelian Fields....Pages 409-483
Factorizations....Pages 485-510
Back Matter....Pages 511-710


This book gives an exposition of the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. The following topics are treated: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. Each chapter ends with exercises and a short review of the relevant literature up to 2003. The bibliography  has over 3400 items.


Content:
Front Matter....Pages I-XI
Dedekind Domains and Valuations....Pages 1-41
Algebraic Numbers and Integers....Pages 43-83
Units and Ideal Classes....Pages 85-133
Extensions....Pages 135-197
P-adic Fields....Pages 199-255
Applications of the Theory of P-adic Fields....Pages 257-312
Analytical Methods....Pages 313-407
Abelian Fields....Pages 409-483
Factorizations....Pages 485-510
Back Matter....Pages 511-710
....
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