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.
A History of Abstract Algebra
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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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Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century algebra came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the intellectual lineage behind many of the basic concepts, results, and theories of abstract algebra. The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing context from which the reader may gain a deeper appreciation of the mathematics involved. Key features: * Begins with an overview of classical algebra * Contains separate chapters on aspects of the development of groups, rings, and fields * Examines the evolution of linear algebra as it relates to other elements of abstract algebra * Highlights the lives and works of six notables: Cayley, Dedekind, Galois, Gauss, Hamilton, and especially the pioneering work of Emmy Noether * Offers suggestions to instructors on ways of integrating the history of abstract algebra into their teaching * Each chapter concludes with extensive references to the relevant literature Mathematics instructors, algebraists, and historians of science will find the work a valuable reference. The book may also serve as a supplemental text for courses in abstract algebra or the history of mathematics.
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11 of 12 people found the following review helpful:
2.0 out of 5 stars
Shallow, bland and spineless,
This review is from: A History of Abstract Algebra (Paperback)
This book is basically useless. In a thoroughly unoriginal manner, Kleiner recounts the usual textbook story about the development of algebra, but with far too little depth and detail for anyone to gain anything but a smattering of clichés, and with no indication whatever of how this boring encyclopaedic catalog of who said what in what year is supposed to be "useful for teachers of relevant courses [and] for their students" (p. xi), which is the stated goal of the book.The supposed usefulness of history lies in "showing how abstract algebra originated in, and sheds light on, the solution of 'concrete' problems" (p. 103). "History points to the sources of the subject ... It considers the context in which the originator of an idea was working in order to bring to the fore the 'burning problem' which he or she was trying to solve." (p. xii). I could not agree more that history in this sense would be a wonderful resource for teaching and learning. Sadly, however, Kleiner's history is not of this type, his trumpeting notwithstanding. Consider for example the origin of abstract group theory. "In 1854 Caley gave the first abstract definition of a finite group" (p. 31). According to Kleiner, this was "a remarkable accomplishment at this time in the evolution of group theory" (p. 31). How so? What was so "remarkable" about it? What was the "burning problem" that Cayley was trying to solve? None of these questions are answered. Kleiner apparently thinks that no such justification is needed. Because Cayley's paper is agreeable to the modern mathematician it must automatically be "a remarkable accomplishment." In conclusion, this book is less concerned with offering a meaningful complement to the establishment view of abstract algebra than with shamelessly and groundlessly touting the pedigree of the same.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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symbolical algebra, algebraic function theory, abstract group concept, hypercomplex number systems, arithmetical algebra, cyclotomic integers, hypercomplex systems, binary quadratic forms, biquadratic reciprocity, hypercomplex numbers, abstract group theory, classical algebra, noncommutative ring theory, algebraic number fields, algebraic function fields, algebraic number theory Key Phrases - Capitalized Phrases (CAPs): (learn more)
Emmy Noether, Biographies of Selected Mathematicians, Fermat's Last Theorem, William Rowan Hamilton, Evariste Galois, Marcel Dekker, Richard Dedekind, History of Group Theory, Carl Friedrich Gauss, Klein's Erlangen Program, History of Classical Algebra, Arthur Cayley, Dictionary of Scientific Biography, Elements of the History of Mathematics, University of Erlangen, Abstract Algebra Inspired, Charles Scribner's Sons, History of Field Theory, History of Ring Theory, History of Linear Algebra, Men of Mathematics, Fundamental Theorem of Algebra, Dirichlet Principle, Oxford University Press, University of Göttingen Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
symbolical algebra, algebraic function theory, abstract group concept, hypercomplex number systems, arithmetical algebra, cyclotomic integers, hypercomplex systems, binary quadratic forms, biquadratic reciprocity, hypercomplex numbers, abstract group theory, classical algebra, noncommutative ring theory, algebraic number fields, algebraic function fields, algebraic number theory Key Phrases - Capitalized Phrases (CAPs): (learn more)
Emmy Noether, Biographies of Selected Mathematicians, Fermat's Last Theorem, William Rowan Hamilton, Evariste Galois, Marcel Dekker, Richard Dedekind, History of Group Theory, Carl Friedrich Gauss, Klein's Erlangen Program, History of Classical Algebra, Arthur Cayley, Dictionary of Scientific Biography, Elements of the History of Mathematics, University of Erlangen, Abstract Algebra Inspired, Charles Scribner's Sons, History of Field Theory, History of Ring Theory, History of Linear Algebra, Men of Mathematics, Fundamental Theorem of Algebra, Dirichlet Principle, Oxford University Press, University of Göttingen Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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