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A History of Abstract Algebra

From Algebraic Equations to Modern Algebra

  • Jeremy Gray

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
  18. Jeremy Gray
    Pages 217-229
  19. Jeremy Gray
    Pages 231-233
  20. Jeremy Gray
    Pages 245-253
  21. Jeremy Gray
    Pages 255-262
  22. Jeremy Gray
    Pages 263-273
  23. Jeremy Gray
    Pages 275-280
  24. Jeremy Gray
    Pages 281-288
  25. Jeremy Gray
    Pages 289-295
  26. Jeremy Gray
    Pages 297-303
  27. Jeremy Gray
    Pages 305-308
  28. Back Matter
    Pages 309-415

About this book

Introduction

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. 

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

Authors and affiliations

  • Jeremy Gray
    • 1
  1. 1.School of Mathematics and StatisticsThe Open UniversityMilton KeynesUnited Kingdom

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-94773-0
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-94772-3
  • Online ISBN 978-3-319-94773-0
  • Series Print ISSN 1615-2085
  • Series Online ISSN 2197-4144
  • Buy this book on publisher's site