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

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. Front Matter ....Pages i-xxiv Simple Quadratic Forms (Jeremy Gray)....Pages 1-13 Fermat’s Last Theorem (Jeremy Gray)....Pages 15-21 Lagrange’s Theory of Quadratic Forms (Jeremy Gray)....Pages 23-36 Gauss’s Disquisitiones Arithmeticae (Jeremy Gray)....Pages 37-47 Cyclotomy (Jeremy Gray)....Pages 49-56 Two of Gauss’s Proofs of Quadratic Reciprocity (Jeremy Gray)....Pages 57-65 Dirichlet’s Lectures on Quadratic Forms (Jeremy Gray)....Pages 67-77 Is the Quintic Unsolvable? (Jeremy Gray)....Pages 79-95 The Unsolvability of the Quintic (Jeremy Gray)....Pages 97-114 Galois’s Theory (Jeremy Gray)....Pages 115-131 After Galois (Jeremy Gray)....Pages 133-142 Revision and First Assignment (Jeremy Gray)....Pages 143-147 Jordan’s Traité (Jeremy Gray)....Pages 149-161 The Galois Theory of Hermite, Jordan and Klein (Jeremy Gray)....Pages 163-177 What Is ‘Galois Theory’? (Jeremy Gray)....Pages 179-187 Algebraic Number Theory: Cyclotomy (Jeremy Gray)....Pages 189-193 Dedekind’s First Theory of Ideals (Jeremy Gray)....Pages 195-201 Dedekind’s Later Theory of Ideals (Jeremy Gray)....Pages 203-208 Quadratic Forms and Ideals (Jeremy Gray)....Pages 209-215 Kronecker’s Algebraic Number Theory (Jeremy Gray)....Pages 217-229 Revision and Second Assignment (Jeremy Gray)....Pages 231-233 Algebra at the End of the Nineteenth Century (Jeremy Gray)....Pages 235-243 The Concept of an Abstract Field (Jeremy Gray)....Pages 245-253 Ideal Theory and Algebraic Curves (Jeremy Gray)....Pages 255-262 Invariant Theory and Polynomial Rings (Jeremy Gray)....Pages 263-273 Hilbert’s Zahlbericht (Jeremy Gray)....Pages 275-280 The Rise of Modern Algebra: Group Theory (Jeremy Gray)....Pages 281-288 Emmy Noether (Jeremy Gray)....Pages 289-295 From Weber to van der Waerden (Jeremy Gray)....Pages 297-303 Revision and Final Assignment (Jeremy Gray)....Pages 305-308 Back Matter ....Pages 309-415