Product Description

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 history 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.

Customer Reviews (1)

Shallow, bland and spineless
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. ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1601 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

Product Description
This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1650 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more