Class field theory / Nancy Childress.

Format:

• Book

Author/Creator:

• Childress, Nancy, 1958-

Series:

• Universitext

Language:

English

Subjects (All):

• Class field theory.

Physical Description:

x, 226 pages : illustrations ; 24 cm.

Place of Publication:

New York : Springer, [2009]

Summary:

• Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. It brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.
• This book is an accessible introduction to class field theory. It takes a traditional approach in that it attempts to present the material using the original techniques of proof (global to local), but in a fashion which is cleaner and more streamlined than most other books on this topic. It could be used for a graduate course on algebraic number theory, as well as for students who are interested in self-study. The book has been class-tested, and the author has included exercises throughout the text.

Contents:

• 1 A Brief Review 1
• 1 Number Fields 1
• 2 Completions of Number Fields 8
• 3 Some General Questions Motivating Class Field Theory 14
• 2 Dirichlet's Theorem on Primes in Arithmetic Progressions 17
• 1 Characters of Finite Abelian Groups 17
• 2 Dirichlet Characters 20
• 3 Dirichlet Series 30
• 4 Dirichlet's Theorem on Primes in Arithmetic Progressions 35
• 5 Dirichlet Density 40
• 3 Ray Class Groups 45
• 1 The Approximation Theorem and Infinite Primes 45
• 2 Ray Class Groups and the Universal Norm Index Inequality 47
• 3 The Main Theorems of Class Field Theory 60
• 4 The Idelic Theory 63
• 1 Places of a Number Field 64
• 2 A Little Topology 66
• 3 The Group of Ideles of a Number Field 68
• 4 Cohomology of Finite Cyclic Groups and the Herbrand Quotient 75
• 5 Cyclic Galois Action on Ideles 83
• 5 Artin Reciprocity 105
• 1 The Conductor of an Abelian Extension of Number Fields and the Artin Symbol 105
• 2 Artin Reciprocity 111
• 3 An Example: Quadratic Reciprocity 128
• 4 Some Preliminary Results about the Artin Map on Local Fields 130
• 6 The Existence Theorem, Consequences and Applications 135
• 1 The Ordering Theorem and the Reduction Lemma 136
• 2 Kummer n-extensions and the Proof of the Existence Theorem 139
• 3 The Artin Map on Local Fields 148
• 4 The Hilbert Class Field 153
• 5 Arbitrary Finite Extensions of Number Fields 159
• 6 Infinite Extensions and an Alternate Proof of the Existence Theorem 162
• 7 An Example: Cyclotomic Fields 168
• 7 Local Class Field Theory 181
• 1 Some Preliminary Facts About Local Fields 182
• 2 A Fundamental Exact Sequence 186
• 3 Local Units Modulo Norms 191
• 4 One-Dimensional Formal Group Laws 195
• 5 The Formal Group Laws of Lubin and Tate 198
• 6 Lubin-Tate Extensions 201
• 7 The Local Artin Map 210.

Notes:

Includes bibliographical references (pages 219-222) and index.

ISBN:

• 9780387724898
• 0387724893

OCLC:

163616596