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

## Book Description

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

## Book Details

- Title: A History of Abstract Algebra: From Algebraic Equations to Modern Algebra
- Author: Jeremy Gray
- Length: 415 pages
- Edition: 1st ed. 2018
- Language: English
- Publisher: Springer
- Publication Date: 2018-09-12
- ISBN-10: 3319947729
- ISBN-13: 9783319947723

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