A History of Abstract Algebra
From Algebraic Equations to Modern Algebra
Authors: Gray, Jeremy J.
Free Preview Provides a wideranging and uptodate account on the history of abstract algebra
 Covers topics from number theory (especially quadratic forms) and Galois theory as far as the origins of the abstract theories of groups, rings and fields
 Develops the mathematical and the historical skills needed to understand the subject
 Presents material that is difficult to find elsewhere, including translations of Gauss’s sixth proof of quadratic reciprocity, parts of Jordan’s Traité and Dedekind’s 11th supplement, as well as a summary of Klein’s work on the icosahedron
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 About this Textbook

This textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject.
Beginning with Gauss’s theory of numbers and Galois’s ideas, the book progresses to Dedekind and Kronecker, Jordan and Klein, Steinitz, Hilbert, and Emmy Noether. Approaching mathematical topics from a historical perspective, the author explores quadratic forms, quadratic reciprocity, Fermat’s Last Theorem, cyclotomy, quintic equations, Galois theory, commutative rings, abstract fields, ideal theory, invariant theory, and group theory. Readers will learn what Galois accomplished, how difficult the proofs of his theorems were, and how important Camille Jordan and Felix Klein were in the eventual acceptance of Galois’s approach to the solution of equations. The book also describes the relationship between Kummer’s ideal numbers and Dedekind’s ideals, and discusses why Dedekind felt his solution to the divisor problem was better than Kummer’s.Designed for a course in the history of modern algebra, this book is aimed at undergraduate students with an introductory background in algebra but will also appeal to researchers with a general interest in the topic. With exercises at the end of each chapter and appendices providing material difficult to find elsewhere, this book is selfcontained and therefore suitable for selfstudy.
 About the authors

Jeremy Gray is a leading historian of modern mathematics. He has been awarded the Leon Whiteman Prize of the American Mathematical Society and the Neugebauer Prize of the European Mathematical Society for his work, and is a Fellow of the American Mathematical Society.
 Reviews

“The book under review is an excellent contribution to the history of abstract algebra and the beginnings of algebraic number theory. I recommend it to everyone interested in the history of mathematics.” (Franz Lemmermeyer, zbMATH 1411.01005, 2019)
“This is a nice book to have around; it reflects careful scholarship and is filled with interesting material. … there is much to like about this book. It is quite detailed, contains a lot of information, is meticulously researched, and has an extensive bibliography. Anyone interested in the history of mathematics, or abstract algebra, will want to make the acquaintance of this book.” (Mark Hunacek, MAA Reviews, June 24, 2019)
 Table of contents (30 chapters)


Simple Quadratic Forms
Pages 113

Fermat’s Last Theorem
Pages 1521

Lagrange’s Theory of Quadratic Forms
Pages 2336

Gauss’s Disquisitiones Arithmeticae
Pages 3747

Cyclotomy
Pages 4956

Table of contents (30 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 A History of Abstract Algebra
 Book Subtitle
 From Algebraic Equations to Modern Algebra
 Authors

 Jeremy J. Gray
 Series Title
 Springer Undergraduate Mathematics Series
 Copyright
 2018
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer Nature Switzerland AG
 eBook ISBN
 9783319947730
 DOI
 10.1007/9783319947730
 Softcover ISBN
 9783319947723
 Series ISSN
 16152085
 Edition Number
 1
 Number of Pages
 XXIV, 415
 Number of Illustrations
 18 b/w illustrations
 Topics