A History of Abstract Algebra: From Algebraic Equations to Modern Algebra

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.

Table of Contents

Chapter 1 Simple Quadratic Forms
Chapter 2 Fermat’S Last Theorem
Chapter 3 Lagrange’S Theory Of Quadratic Forms
Chapter 4 Gauss’S Disquisitiones Arithmeticae
Chapter 5 Cyclotomy
Chapter 6 Two Of Gauss’S Proofs Of Quadratic Reciprocity
Chapter 7 Dirichlet’S Lectures On Quadratic Forms
Chapter 8 Is The Quintic Unsolvable?
Chapter 9 The Unsolvability Of The Quintic
Chapter 10 Galois’S Theory
Chapter 11 After Galois
Chapter 12 Revision And First Assignment
Chapter 13 Jordan’S Traité
Chapter 14 The Galois Theory Of Hermite, Jordan And Klein
Chapter 15 What Is `Galois Theory’?
Chapter 16 Algebraic Number Theory: Cyclotomy
Chapter 17 Dedekind’S First Theory Of Ideals
Chapter 18 Dedekind’S Later Theory Of Ideals
Chapter 19 Quadratic Forms And Ideals
Chapter 20 Kronecker’S Algebraic Number Theory
Chapter 21 Revision And Second Assignment
Chapter 22 Algebra At The End Of The Nineteenth Century
Chapter 23 The Concept Of An Abstract Field
Chapter 24 Ideal Theory And Algebraic Curves
Chapter 25 Invariant Theory And Polynomial Rings
Chapter 26 Hilbert’S Zahlbericht
Chapter 27 The Rise Of Modern Algebra: Group Theory
Chapter 28 Emmy Noether
Chapter 29 From Weber To Van Der Waerden
Chapter 30 Revision And Final Assignment
Appendix A Polynomial Equations In The Eighteenth Century
Appendix B Gauss And Composition Of Forms
Appendix C Gauss’S Fourth And Sixth Proofs Of Quadratic Reciprocity
Appendix D From Jordan’S Traité
Appendix E Klein’S Erlanger Programm, Groups And Geometry
Appendix F From Dedekind’S 11Th Supplement (1894)
Appendix G Subgroups Of S4 And S5
Appendix H Curves And Projective Space
Appendix I Resultants
Appendix J Further Reading