Primes Of The Form X2 Ny2

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Primes of the Form x2 + ny2

David A. Cox
Primes of the Form x2 ny2 Book PDF

Author : David A. Cox
Publisher : John Wiley & Sons
Page : 372 pages
File Size : 47,8 Mb
Release : 2011-10-24
Category : Mathematics
ISBN : 9781118031001

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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.

Primes of the Form x2+ny2

David A. Cox
Primes of the Form x2 ny2 Book PDF

Author : David A. Cox
Publisher : John Wiley & Sons
Page : 363 pages
File Size : 50,6 Mb
Release : 2014-08-21
Category : Mathematics
ISBN : 9781118400746

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

An exciting approach to the history and mathematics of number theory “. . . the author’s style is totally lucid and very easy to read . . .the result is indeed a wonderful story.” —Mathematical Reviews Written in a unique and accessible style for readers of varied mathematical backgrounds, the Second Edition of Primes of the Form p = x2+ ny2 details the history behind how Pierre de Fermat’s work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. The book also illustrates how results of Euler and Gauss can be fully understood only in the context of class field theory, and in addition, explores a selection of the magnificent formulas of complex multiplication. Primes of the Form p = x2 + ny2, Second Edition focuses on addressing the question of when a prime p is of the form x2 + ny2, which serves as the basis for further discussion of various mathematical topics. This updated edition has several new notable features, including: • A well-motivated introduction to the classical formulation of class field theory • Illustrations of explicit numerical examples to demonstrate the power of basic theorems in various situations • An elementary treatment of quadratic forms and genus theory • Simultaneous treatment of elementary and advanced aspects of number theory • New coverage of the Shimura reciprocity law and a selection of recent work in an updated bibliography Primes of the Form p = x2 + ny2, Second Edition is both a useful reference for number theory theorists and an excellent text for undergraduate and graduate-level courses in number and Galois theory.

Primes of the Form X2 + Ny2

David A. Cox
Primes of the Form X2 Ny2 Book PDF

Author : David A. Cox
Publisher : Wiley-Interscience
Page : 380 pages
File Size : 48,9 Mb
Release : 1989-09-28
Category : Mathematics
ISBN : UOM:39076001018543

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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.

Galois Theory

David A. Cox
Galois Theory Book PDF

Author : David A. Cox
Publisher : John Wiley & Sons
Page : 602 pages
File Size : 48,7 Mb
Release : 2012-03-27
Category : Mathematics
ISBN : 9781118218426

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Galois Theory by David A. Cox Pdf

Praise for the First Edition ". . .will certainly fascinate anyone interested in abstractalgebra: a remarkable book!" —Monatshefte fur Mathematik Galois theory is one of the most established topics inmathematics, with historical roots that led to the development ofmany central concepts in modern algebra, including groups andfields. Covering classic applications of the theory, such assolvability by radicals, geometric constructions, and finitefields, Galois Theory, Second Edition delves into noveltopics like Abel’s theory of Abelian equations, casusirreducibili, and the Galois theory of origami. In addition, this book features detailed treatments of severaltopics not covered in standard texts on Galois theory,including: The contributions of Lagrange, Galois, and Kronecker How to compute Galois groups Galois's results about irreducible polynomials of primeor prime-squared degree Abel's theorem about geometric constructions on thelemniscates Galois groups of quartic polynomials in allcharacteristics Throughout the book, intriguing Mathematical Notes andHistorical Notes sections clarify the discussed ideas andthe historical context; numerous exercises and examples use Mapleand Mathematica to showcase the computations related to Galoistheory; and extensive references have been added to provide readerswith additional resources for further study. Galois Theory, Second Edition is an excellent book forcourses on abstract algebra at the upper-undergraduate and graduatelevels. The book also serves as an interesting reference for anyonewith a general interest in Galois theory and its contributions tothe field of mathematics.

Number Theory

Anonim
Number Theory Book PDF

Author : Anonim
Publisher : Academic Press
Page : 449 pages
File Size : 45,8 Mb
Release : 1986-05-05
Category : Mathematics
ISBN : 9780080873329

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Number Theory by Anonim Pdf

This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them. Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences.

A Course in Computational Algebraic Number Theory

Henri Cohen
A Course in Computational Algebraic Number Theory Book PDF

Author : Henri Cohen
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 48,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662029459

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A Course in Computational Algebraic Number Theory by Henri Cohen Pdf

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

Advanced Topics in Computational Number Theory

Henri Cohen
Advanced Topics in Computational Number Theory Book PDF

Author : Henri Cohen
Publisher : Springer Science & Business Media
Page : 591 pages
File Size : 50,9 Mb
Release : 2012-10-29
Category : Mathematics
ISBN : 9781441984890

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Advanced Topics in Computational Number Theory by Henri Cohen Pdf

Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.

Primes of the Form x2+ny2 : Fermat, Class Field Theory, and Complex Multiplication. Third Edition with Solutions

David A. Cox
Primes of the Form x2 ny2 Fermat Class Field Theory and Complex Multiplication Third Edition with Solutions Book PDF

Author : David A. Cox
Publisher : American Mathematical Soc.
Page : 551 pages
File Size : 51,6 Mb
Release : 2022-11-16
Category : Education
ISBN : 9781470470289

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Primes of the Form x2+ny2 : Fermat, Class Field Theory, and Complex Multiplication. Third Edition with Solutions by David A. Cox Pdf

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.

A Primer of Analytic Number Theory

Jeffrey Stopple
A Primer of Analytic Number Theory Book PDF

Author : Jeffrey Stopple
Publisher : Cambridge University Press
Page : 404 pages
File Size : 42,9 Mb
Release : 2003-06-23
Category : Mathematics
ISBN : 0521012538

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A Primer of Analytic Number Theory by Jeffrey Stopple Pdf

An undergraduate-level 2003 introduction whose only prerequisite is a standard calculus course.

My Numbers, My Friends

Paulo Ribenboim
My Numbers My Friends Book PDF

Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
Page : 384 pages
File Size : 46,8 Mb
Release : 2006-05-10
Category : Mathematics
ISBN : 9780387227542

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My Numbers, My Friends by Paulo Ribenboim Pdf

This selection of expository essays by Paulo Ribenboim should be of interest to mathematicians from all walks. Ribenboim, a highly praised author of several popular titles, writes each essay in a light and humorous language without secrets, making them thoroughly accessible to everyone with an interest in numbers. This new collection includes essays on Fibonacci numbers, prime numbers, Bernoulli numbers, and historical presentations of the main problems pertaining to elementary number theory, such as Kummers work on Fermat's last theorem.

Introduction to the Arithmetic Theory of Automorphic Functions

Gorō Shimura
Introduction to the Arithmetic Theory of Automorphic Functions Book PDF

Author : Gorō Shimura
Publisher : Princeton University Press
Page : 292 pages
File Size : 49,7 Mb
Release : 1971-08-21
Category : Mathematics
ISBN : 0691080925

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Introduction to the Arithmetic Theory of Automorphic Functions by Gorō Shimura Pdf

The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.

Disquisitiones Arithmeticae

Carl Friedrich Gauss,William C. Waterhouse
Disquisitiones Arithmeticae Book PDF

Author : Carl Friedrich Gauss,William C. Waterhouse
Publisher : Springer
Page : 472 pages
File Size : 52,6 Mb
Release : 2018-02-07
Category : Mathematics
ISBN : 9781493975600

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Disquisitiones Arithmeticae by Carl Friedrich Gauss,William C. Waterhouse Pdf

Carl Friedrich Gauss’s textbook, Disquisitiones arithmeticae, published in 1801 (Latin), remains to this day a true masterpiece of mathematical examination. .

Modular Forms and Fermat’s Last Theorem

Gary Cornell,Joseph H. Silverman,Glenn Stevens
Modular Forms and Fermat s Last Theorem Book PDF

Author : Gary Cornell,Joseph H. Silverman,Glenn Stevens
Publisher : Springer Science & Business Media
Page : 592 pages
File Size : 40,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461219743

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Modular Forms and Fermat’s Last Theorem by Gary Cornell,Joseph H. Silverman,Glenn Stevens Pdf

This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

Solved and Unsolved Problems in Number Theory

Daniel Shanks
Solved and Unsolved Problems in Number Theory Book PDF

Author : Daniel Shanks
Publisher : American Mathematical Society
Page : 321 pages
File Size : 53,9 Mb
Release : 2024-01-24
Category : Mathematics
ISBN : 9781470476458

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Solved and Unsolved Problems in Number Theory by Daniel Shanks Pdf

The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.

Knots and Primes

Masanori Morishita
Knots and Primes Book PDF

Author : Masanori Morishita
Publisher : Springer Science & Business Media
Page : 192 pages
File Size : 47,6 Mb
Release : 2011-11-27
Category : Mathematics
ISBN : 9781447121589

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Knots and Primes by Masanori Morishita Pdf

This is a foundation for arithmetic topology - a new branch of mathematics which is focused upon the analogy between knot theory and number theory. Starting with an informative introduction to its origins, namely Gauss, this text provides a background on knots, three manifolds and number fields. Common aspects of both knot theory and number theory, for instance knots in three manifolds versus primes in a number field, are compared throughout the book. These comparisons begin at an elementary level, slowly building up to advanced theories in later chapters. Definitions are carefully formulated and proofs are largely self-contained. When necessary, background information is provided and theory is accompanied with a number of useful examples and illustrations, making this a useful text for both undergraduates and graduates in the field of knot theory, number theory and geometry. ​