Class Field Theory

Author : Nancy Childress
Publisher : Springer Science & Business Media
Page : 230 pages
File Size : 40,9 Mb
Release : 2008-10-28
Category : Mathematics
ISBN : 9780387724904

Get Book

Class Field Theory by Nancy Childress Pdf

Class field theory brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. This book provides 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, 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 lots of challenging exercises throughout the text.

Author : Georges Gras
Publisher : Springer Science & Business Media
Page : 491 pages
File Size : 42,9 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662113233

Get Book

Class Field Theory by Georges Gras Pdf

Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book.

Author : J. Neukirch
Publisher : Springer Science & Business Media
Page : 148 pages
File Size : 43,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642824654

Get Book

Class Field Theory by J. Neukirch Pdf

Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group theoretical principle, admitting an entirely elementary description. This de scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.

Author : David Harari
Publisher : Springer Nature
Page : 336 pages
File Size : 41,7 Mb
Release : 2020-06-24
Category : Mathematics
ISBN : 9783030439019

Get Book

Galois Cohomology and Class Field Theory by David Harari Pdf

This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory. Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 211 pages
File Size : 45,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461210351

Get Book

Algebraic Groups and Class Fields by Jean-Pierre Serre Pdf

Translation of the French Edition

Author : Pierre Guillot
Publisher : Cambridge University Press
Page : 309 pages
File Size : 47,6 Mb
Release : 2018-11
Category : Mathematics
ISBN : 9781108421775

Get Book

A Gentle Course in Local Class Field Theory by Pierre Guillot Pdf

A self-contained exposition of local class field theory for students in advanced algebra.

Author : David A. Cox
Publisher : John Wiley & Sons
Page : 372 pages
File Size : 42,8 Mb
Release : 2011-10-24
Category : Mathematics
ISBN : 9781118031001

Get Book

Primes of the Form x2 + ny2 by David A. Cox Pdf

Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of field theory and its intimate connection with complex multiplication. While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included. The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively.

Author : Jürgen Neukirch
Publisher : Springer Science & Business Media
Page : 195 pages
File Size : 54,8 Mb
Release : 2013-04-08
Category : Mathematics
ISBN : 9783642354373

Get Book

Class Field Theory by Jürgen Neukirch Pdf

The present manuscript is an improved edition of a text that first appeared under the same title in Bonner Mathematische Schriften, no.26, and originated from a series of lectures given by the author in 1965/66 in Wolfgang Krull's seminar in Bonn. Its main goal is to provide the reader, acquainted with the basics of algebraic number theory, a quick and immediate access to class field theory. This script consists of three parts, the first of which discusses the cohomology of finite groups. The second part discusses local class field theory, and the third part concerns the class field theory of finite algebraic number fields.

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 249 pages
File Size : 49,7 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475756739

Get Book

Local Fields by Jean-Pierre Serre Pdf

The goal of this book is to present local class field theory from the cohomo logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray.

Author : Kenkichi Iwasawa
Publisher : Oxford University Press, USA
Page : 184 pages
File Size : 49,6 Mb
Release : 1986
Category : History
ISBN : UOM:39015015612537

Get Book

Local Class Field Theory by Kenkichi Iwasawa Pdf

This readable introduction to local class field theory, a theory of algebraic extensions, covers such topics as abelian extensions. Almost self-contained, the book is accessible to any reader with a basic background in algebra and topological groups.

Author : Harvey Cohn
Publisher : Springer Science & Business Media
Page : 344 pages
File Size : 51,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461299509

Get Book

A Classical Invitation to Algebraic Numbers and Class Fields by Harvey Cohn Pdf

"Artin's 1932 Göttingen Lectures on Class Field Theory" and "Connections between Algebrac Number Theory and Integral Matrices"

Author : Toyokazu Hiramatsu,Seiken Saito
Publisher : World Scientific
Page : 188 pages
File Size : 49,8 Mb
Release : 2016-09-13
Category : Mathematics
ISBN : 9789813142282

Get Book

An Introduction to Non-Abelian Class Field Theory by Toyokazu Hiramatsu,Seiken Saito Pdf

This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.

Author : Emil Artin,John Torrence Tate,J. Tate
Publisher : Unknown
Page : 296 pages
File Size : 47,7 Mb
Release : 1967
Category : Algebra
ISBN : UCSC:32106002310784

Get Book

Class Field Theory by Emil Artin,John Torrence Tate,J. Tate Pdf

Author : Harvey Cohn
Publisher : Courier Corporation
Page : 244 pages
File Size : 55,7 Mb
Release : 1994-01-01
Category : Mathematics
ISBN : 048668346X

Get Book

Introduction to the Construction of Class Fields by Harvey Cohn Pdf

A broad introduction to quadratic forms, modular functions, interpretation by rings and ideals, class fields by radicals and more. 1985 ed.

Author : Daniel A. Marcus
Publisher : Springer
Page : 203 pages
File Size : 53,7 Mb
Release : 2018-07-05
Category : Mathematics
ISBN : 9783319902333

Get Book

Number Fields by Daniel A. Marcus Pdf

Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.