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OverviewCareful organisation and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract techniques constitute the primary focus. Topics include introductory materials on elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.Subjects correspond to those usually covered in a one-semester, graduate level course in algebraic number theory, making this book ideal either for classroom use or as a stimulating series of exercises for mathematically minded individuals. Full Product DetailsAuthor: A. WeissPublisher: Dover Publications Inc. Imprint: Dover Publications Inc. Dimensions: Width: 14.00cm , Height: 1.60cm , Length: 21.60cm Weight: 0.318kg ISBN: 9780486401898ISBN 10: 0486401898 Pages: 288 Publication Date: 29 January 1998 Audience: College/higher education , Professional and scholarly , Undergraduate , Professional & Vocational Format: Paperback Publisher's Status: Out of Print Availability: Out of stock  Table of ContentsPreface; References Chapter 1. Elementary Valuation Theory 1.1 Valuations and Prime Divisors 1.2 The Approximation Theorem 1.3 Archimedean and Nonarchimedean Prime Divisors 1.4 The Prime Divisors of Q 1.5 Fields with a Discrete Prime Divisor 1.6 e and f 1.7 Completions 1.8 The Theorem of Ostrowski 1.9 Complete Fields with Discrete Prime Divisor; Exercises Chapter 2. Extension of Valuations 2.1 Uniqueness of Extensions (Complete Case) 2.2 Existence of Extensions (Complete Case) 2.3 Extensions of Discrete Prime Divisors 2.4 Extensions in the General Case 2.5 Consequences; Exercises Chapter 3. Local Fields 3.1 Newton's Method 3.2 Unramified Extensions 3.3 Totally Ramified Extensions 3.4 Tamely Ramified Extensions 3.5 Inertia Group 3.6 Ramification Groups 3.7 Different and Discriminant; Exercises Chapter 4. Ordinary Arithmetic Fields 4.1 Axioms and Basic Properties 4.2 Ideals and Divisors 4.3 The Fundamental Theorem of OAFs 4.4 Dedekind Rings 4.5 Over-rings of O 4.6 Class Number 4.7 Mappings of Ideals 4.8 Different and Discriminant 4.9 Factoring Prime Ideals in an Extension Field 4.10 Hilbert Theory; Exercises Chapter 5. Global Fields 5.1 Global Fields and the Product Formula 5.2 Adeles, Ideles, Divisors, and Ideals 5.3 Unit Theorem and Class Number 5.4 Class Number of an Algebraic Number Field 5.5 Topological Considerations 5.6 Relative Theory; Exercises Chapter 6. Quadratic Fields 6.1 Integral Basis and Discriminant 6.2 Prime Ideals 6.3 Units 6.4 Class Number 6.5 The Local Situation 6.6 Norm Residue Symbol Chapter 7. Cyclotomic Fields 7.1 Elementary Facts 7.2 Unramified Primes 7.3 Quadratic Reciprocity Law 7.4 Ramified Primes 7.5 Integral Basis and Discriminant 7.6 Units 7.7 Class Number Symbols and Notation; IndexReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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