Using Algebraic Geometry (Graduate Texts in Mathematics, 185)

The discovery of new algorithms for dealing with polynomial equations, and their implementation on fast, inexpensive computers, has revolutionized algebraic geometry and led to exciting new applications in the field. This book details many uses of algebraic geometry and highlights recent applications of Grobner bases and resultants. This edition contains two new sections, a new chapter, updated references and many minor improvements throughout.

Computers & Technology

Cox, David A.

O'Shea, Donal

Little,John

Computer Science

Springer

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

Science & Math

Mathematics

Springer, Verlag

This set features: Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication (978-0-471-19079-0) and Galois Theory (978-0-471-43419-1) both by David A. Cox.

The Collected Works of David A. Cox (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

wiley

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: Fermat, Class Field Theory, and Complex Multiplication

Wiley-Interscience

An introduction to one of the most celebrated theories of mathematics 
Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox’s Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics. Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel’s theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami. Anyone fascinated by abstract algebra will find careful discussions of such topics as: <ul>
<li>The contributions of Lagrange, Galois, and Kronecker</li> <li>How to compute Galois groups</li> <li>Galois’s results about irreducible polynomials of prime or prime-squared degree</li> <li>Abel’s theorem about geometric constructions on the lemniscate</li> </ul>
With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike.

Galois Theory

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: Fermat, Class Field Theory, and Complex Multiplication (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

For courses currently engaged, or leaning toward calculus reform. Callahan fully embraces the calculus reform movement in technology and pedagogy, while taking it a step further with a unique organization and applications to real-world problems.

Calculus in Context

CALLAHAN, James

Hoffman, Kenneth R.

Pollatsek, Harriet

Senechal, Lester

W.H.Freeman

Book by Cox, David A., Katz, Sheldon

Mirror Symmetry and Algebraic Geometry (Mathematical Surveys & Monographs)

Katz, Sheldon

Amer Mathematical Society

Praise for the First Edition ". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!" ―Monatshefte fur Mathematik Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel’s theory of Abelian equations, casus irreducibili, and the Galois theory of origami. In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including: <ul>
<li>The contributions of Lagrange, Galois, and Kronecker</li> <li>How to compute Galois groups</li> <li>Galois's results about irreducible polynomials of prime or prime-squared degree</li> <li>Abel's theorem about geometric constructions on the lemniscates</li> <li>Galois groups of quartic polynomials in all characteristics</li> </ul>
Throughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study. Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.

See the best price to sell, buy, or rent books by Cox, David A.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Cox, David A.

See the best price to sell, buy, or rent books by Cox, David A.. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Using Algebraic Geometry (Graduate Texts in Mathematics, 185)

Using Algebraic Geometry (Graduate Texts in Mathematics, 185)

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

The Collected Works of David A. Cox (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

Galois Theory

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

Calculus in Context

Mirror Symmetry and Algebraic Geometry (Mathematical Surveys & Monographs)

Galois Theory