© 2018

A History of Abstract Algebra

From Algebraic Equations to Modern Algebra


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Jeremy Gray
    Pages 1-13
  3. Jeremy Gray
    Pages 15-21
  4. Jeremy Gray
    Pages 23-36
  5. Jeremy Gray
    Pages 37-47
  6. Jeremy Gray
    Pages 49-56
  7. Jeremy Gray
    Pages 79-95
  8. Jeremy Gray
    Pages 97-114
  9. Jeremy Gray
    Pages 115-131
  10. Jeremy Gray
    Pages 133-142
  11. Jeremy Gray
    Pages 143-147
  12. Jeremy Gray
    Pages 149-161
  13. Jeremy Gray
    Pages 179-187
  14. Jeremy Gray
    Pages 189-193
  15. Jeremy Gray
    Pages 195-201
  16. Jeremy Gray
    Pages 203-208
  17. Jeremy Gray
    Pages 209-215

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 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 self-contained and therefore suitable for self-study. 


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

Authors and affiliations

  1. 1.School of Mathematics and StatisticsThe Open UniversityMilton KeynesUnited Kingdom

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.

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
  • Series Abbreviated Title SUMS
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-94772-3
  • eBook ISBN 978-3-319-94773-0
  • Series ISSN 1615-2085
  • Series E-ISSN 2197-4144
  • Edition Number 1
  • Number of Pages XXIV, 415
  • Number of Illustrations 18 b/w illustrations, 0 illustrations in colour
  • Topics History of Mathematical Sciences
    Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


“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)