Author : Harvey Cohn
Publisher : Courier Corporation
Page : 244 pages
File Size : 33,37 MB
Release : 1994-01-01
Category : Mathematics
ISBN : 9780486683461
A broad introduction to quadratic forms, modular functions, interpretation by rings and ideals, class fields by radicals and more. 1985 ed.
Author : Harvey Cohn
Publisher : CUP Archive
Page : 232 pages
File Size : 50,40 MB
Release : 1985-08-30
Category : Mathematics
ISBN : 9780521247627
In this graduate level textbook, Professor Cohn takes a problem that Pythagoras could have posed, and using it as motivation, develops a constructional introduction to classical field theory and modular function theory. The interest in constructional techniques has increased recently with the advent of cheap and plentiful computer technology. The beginning chapters provide the motivation and necessary background in elementary algebraic number theory and Riemann surface theory. The ideas and results are then applied and extended to class field theory. In the later chapters, more specialized results are presented, with full proofs, though the author emphasizes, with examples, the relation of the material to other parts of mathematics.
Author : David A. Cox
Publisher : American Mathematical Soc.
Page : 551 pages
File Size : 17,24 MB
Release : 2022-11-16
Category : Education
ISBN : 1470470284
This book studies when a prime p can be written in the form x2+ny2. It begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. After exploring cubic and biquadratic reciprocity, the pace quickens with the introduction of algebraic number fields and class field theory. This leads to the concept of ring class field and a complete but abstract solution of p=x2+ny2. To make things more concrete, the book introduces complex multiplication and modular functions to give a constructive solution. The book ends with a discussion of elliptic curves and Shimura reciprocity. Along the way the reader will encounter some compelling history and marvelous formulas, together with a complete solution of the class number one problem for imaginary quadratic fields. The book is accessible to readers with modest backgrounds in number theory. In the third edition, the numerous exercises have been thoroughly checked and revised, and as a special feature, complete solutions are included. This makes the book especially attractive to readers who want to get an active knowledge of this wonderful part of mathematics.
Author : Wladyslaw Narkiewicz
Publisher : Springer Science & Business Media
Page : 712 pages
File Size : 12,64 MB
Release : 2013-06-29
Category : Mathematics
ISBN : 3662070014
This book details the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. Coverage includes: 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. The book also features exercises and a list of open problems.
Author : Georges Gras
Publisher : Springer Science & Business Media
Page : 517 pages
File Size : 22,78 MB
Release : 2013-11-11
Category : Mathematics
ISBN : 3662113236
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 : David A. Cox
Publisher : John Wiley & Sons
Page : 372 pages
File Size : 47,43 MB
Release : 2011-10-24
Category : Mathematics
ISBN : 1118031008
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 : David V. Chudnovsky
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 36,94 MB
Release : 2013-11-21
Category : Mathematics
ISBN : 1475741588
New York Number Theory Seminar started its regular meeting in January, 1982. The Seminar has been meeting on a regular basis weekly during the academic year since then. The meeting place of the seminar is in midtown Manhattan at the Graduate School and University Center of the City Uni versity of New York. This central location allows number-theorists in the New York metropolitan area and vistors an easy access. Four volumes of the Seminar proceedings, containing expanded texts of Seminar's lectures had been published in the Springer's Lecture Notes in Mathematics series as volumes 1052 (1984), 1135 (1985), 1240 (1987), and 1383 (1989). Seminar co chairmen are pleased that some of the contributions to the Seminar opened new avenues of research in Number Theory and related areas. On a histori cal note, one of such contributions proved to be a contribution by P. Landweber. In addition to classical and modern Number Theory, this Semi nar encourages Computational Number Theory. This book presents a selection of invited lectures presented at the New York Number Theory Seminar during 1989-1990. These papers cover wide areas of Number Theory, particularly modular functions, Aigebraic and Diophantine Geometry, and Computational Number Theory. The review of C-L. Chai presents a broad view of the moduli of Abelian varieties based on recent work of the author and many other prominent experts. This provides the reader interested in Diophantine Analysis with access to state of the art research. The paper of D. V. and G. V.
Author : Seifallah Randjbar-daemi
Publisher : World Scientific
Page : 322 pages
File Size : 18,80 MB
Release : 1990-06-27
Category :
ISBN : 981461145X
Author :
Publisher :
Page : 704 pages
File Size : 27,14 MB
Release : 1942
Category : Greek letter societies
ISBN :
Author : Hugh C. Williams
Publisher : American Mathematical Soc.
Page : 412 pages
File Size : 44,13 MB
Release :
Category : Science
ISBN : 9780821887592
This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams' sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory.
