Authors:
- Provides a wide-ranging and up-to-date 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
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
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Table of contents (30 chapters)
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Front Matter
About this book
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 relationshipbetween 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 self-contained and therefore suitable for self-study.
Keywords
- MSC (2010): 01A55, 01A60, 01A50, 11-03, 12-03, 13-03
- algebraic number theory
- Galois theory
- quadratic forms
- quadratic reciprocity
- group theory
- commutative rings
- abstract fields
- ideal theory
- Klein Erlangen program
- modern algebra history
- Fermat's Last Theorem
- Cyclotomy
- quintic equation
- Klein’s Icosahedron
- Dedekind theory of ideals
- quadratic forms and ideals
- invariant theory
- Zahlbericht Hilbert
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 tolike 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)
Authors and Affiliations
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School of Mathematics and Statistics, The Open University, Milton Keynes, United Kingdom
Jeremy Gray
About the author
Bibliographic Information
Book Title: A History of Abstract Algebra
Book Subtitle: From Algebraic Equations to Modern Algebra
Authors: Jeremy Gray
Series Title: Springer Undergraduate Mathematics Series
DOI: https://doi.org/10.1007/978-3-319-94773-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Softcover ISBN: 978-3-319-94772-3Published: 16 August 2018
eBook ISBN: 978-3-319-94773-0Published: 07 August 2018
Series ISSN: 1615-2085
Series E-ISSN: 2197-4144
Edition Number: 1
Number of Pages: XXIV, 415
Number of Illustrations: 18 b/w illustrations
Topics: History of Mathematical Sciences, Algebra, Number Theory