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Algebraic Number Theory

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Synopses & Reviews

Publisher Comments:

Careful organization 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.

Synopsis:

Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.

Synopsis:

Careful organization and clear, detailed proofs make this book ideal either for classroom use or as a stimulating series of exercises for mathematically-minded individuals. Modern abstract techniques focus on introducing elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.

Table of Contents

  Preface; 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 Adèles, Idèles, 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; Index

Product Details

ISBN:
9780486401898
Author:
Weiss, Edwin
Publisher:
Dover Publications
Author:
Mathematics
Location:
Mineola, N.Y. :
Subject:
Number Theory
Subject:
Algebraic number theory
Subject:
General Mathematics
Subject:
Mathematics-Number Theory
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
19980131
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
288
Dimensions:
8.5 x 5.38 in 0.71 lb

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Algebraic Number Theory New Trade Paper
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Product details 288 pages Dover Publications - English 9780486401898 Reviews:
"Synopsis" by ,
Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.

"Synopsis" by ,
Careful organization and clear, detailed proofs make this book ideal either for classroom use or as a stimulating series of exercises for mathematically-minded individuals. Modern abstract techniques focus on introducing elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.

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