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This book does nothing less than provide an account of the intellectual lineage of abstract algebra. The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared insoluble by classical means. A major theme of the book is to show how abstract algebra has arisen in attempting to solve some of these classical problems, providing a context from which the reader may gain a deeper appreciation of the mathematics involved. Mathematics instructors, algebraists, and historians of science will find the work a valuable reference.…mehr
This book does nothing less than provide an account of the intellectual lineage of abstract algebra. The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared insoluble by classical means. A major theme of the book is to show how abstract algebra has arisen in attempting to solve some of these classical problems, providing a context from which the reader may gain a deeper appreciation of the mathematics involved. Mathematics instructors, algebraists, and historians of science will find the work a valuable reference.
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Israel Kleiner, York University, Toronto, ON, Canada
Preface.-Chapter 1: Classical Algebra.-Early roots.-The Greeks.-Al-Khwarizmi.-Cubic and quartic equations.-The cubic and complex numbers.-Algebraic notation: Viète and Descartes.-The theory of equations and the Fundamental Theorem of Algebra.-Symbolical algebra.-References.-Chapter 2: Group Theory.-Sources of group theory.-Development of 'specialized' theories of groups.-Emergence of abstraction in group theory.-Consolidation of the abstract group concept; dawn of abstract group theory. Divergence of developments in group theory.-References.-Chapter 3: Ring Theory.-Noncommutative ring theory.-Commutative ring theory.-The abstract definition of a ring.-Emmy Noether and Emil Artin.-Epilogue.-References.-Chapter 4: Field Theory.-Galois theory.-Algebraic number theory.-Algebraic geometry.-Symbolical algebra.-The abstract definition of a field.-Hensel's p-adic numbers.-Steinitz.-A glance ahead.-References.-Chapter 5: Linear Algebra.-Linear equations.-Determinants Matrices and linear transformations.-Linear independence, basis, and dimension.-Vector spaces.-References.-Chapter 6: Emmy Noether and the Advent of Abstract Algebra.-Invariant theory.-Commutative algebra.-Noncommutative algebra and representation theory.-Applications of noncommutative to commutative algebra.-Noether's legacy.-References.-Chapter 7: A course in abstract algebra inspired by history.-Problem I: Why is (-1)(-1) = 1? .-Problem II: What are the integer solutions of x2 + 2 = y3 ? .-Problem III: Can we trisect a 600 angle using only straightedge and compass?.-Problem IV: Can we solve x5 - 6x + 3 = 0? .-Problem V: 'Papa, can you multiply triples?' .-General remarks on the course.-References.-Chapter 8: Biographies of Selected Mathematicians.-Cayley.-Invariants.-Groups.-Matrices. Geometry.-Conclusion.-References.-Dedekind.-Algebraic numbers.-Real numbers.-Natural numbers.-Other works.Conclusion.-References.-Galois.-Mathematics.-Politics.-The duel.-Testament.-Conclusion.-References.-Gauss.-Number theory.-Differential geometry, probability, statistics.-The diary.-Conclusion.-References.-Hamilton.-Optics.-Dynamics.-Complex numbers.-Foundations of algebra.-Quaternions.-Conclusion.-References.-Noether.-Early years.-University studies.-Göttingen.-Noether as a teacher.-Bryn Mawr.-Conclusion.-References.-Index.-Acknowledgments
From the reviews:
"This concise history conveniently brings together topics in modern algebra that one might otherwise only find in scattered sources. ... it reflects a deep attention to the mathematics and to how its history can be used to help understand the subject today. ... The author provides an outline of his course in abstract algebra, a course that is intended for teachers of mathematics ... ." (Albert C. Lewis, Mathematical Reviews, Issue 2008 g)
"This book gives an overview of the origin and development of the basic ideas of modern abstract algebra. ... In each chapter, the author makes extensive references to relevant literature. The book can be recommended to mathematicians, teachers of mathematics (especially of algebra), historians of the sciences and students, who can find many useful references and ideas for their research, teaching or studies." (EMS Newsletter, September, 2008)
"This remarkable book presents both the history of algebra as well as selected detailed biographies of algebraists. ... the origin of important ideas and concepts is presented very skillfully, even in a way such that the development of ideas can be used as a very good textbook for algebra. ... This book combines in relatively few pages non-trivial algebra with detailed historical facts and ideas and should bring the reader a wealth of new insights." (G. Pilz, Internationle Mathematische Nachrichten, Issue 210, 2009)
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