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*A History of Abstract Algebra*, Israel Kleiner, 2007. 168 pages, 24 illustrations, bibliography, $49.95 softcover. ISBN: 978-0-8176-4684-4. Birkhäuser, Boston. www.birkhauser.com

"History points to the sources of abstract algebra, hence to some of its central ideas; it provides motivation; and it makes the subject come to life."

This quote from Kleiner's book captures the essence of his text. The book starts with an introductory chapter on the history of algebra, followed by five chapters that describe the main structures of abstract algebra: groups, rings, fields, and vector spaces. In each chapter we learn about the essential questions, developments and people who created and researched these structures. The last of these five chapters, Emmy Noether and the Advent of Abstract Algebra, is the only chapter devoted to the work of a single mathematician. In it, Kleiner explains why he considers Noether to be the defining figure of abstract algebra.

The final chapter consists of biographical sketches of Cayley, Dedekind, Galois, Gauss, Hamilton, and Emmy Noether. These biographies consist, for the most part, of an explanation of each mathematician's work. They also describe the person's life story and political context, though to a lesser extent. The emphasis is reversed in the case of Galois and Noether. Kleiner's descriptions of these six mathematicians vary in both length and detail. Hamilton, for example, is allotted the most space. Much of his work, even the non-mathematical work, is presented and the discovery of the quaternions is explained in detail. As a consequence, Hamilton's significance as a mathematician appears to be on par with that of Gauss.

Portraits of 24 algebraists are distributed throughout the book. Between the history chapters and the biographies is a short chapter outlining an abstract algebra course inspired by history. This is an interesting idea that will surely appeal to many dedicated teachers and perhaps inspire them to try such an approach. The course is based on five concrete problems from the classical era. Each problem leads to solutions through abstraction and to the development of intricate theories. These theories have multiple "payoffs." That is to say that they go far beyond the intent of the original problems and are often more important than the answers they provide for the motivating questions. The references should be particularly helpful in creating such a course.

The book, according to Kleiner, is written for "teachers of courses in abstract algebra." Some knowledge of basic mathematics is required to follow his arguments. In my own institution, for example, teacher certification requirements motivated my department to include historical perspectives in our abstract algebra (and analysis) course. For a mathematician teaching such a course at an undergraduate level, this book is an ideal resource. Since each algebraic structure has its own chapter, it is possible to read just the corresponding chapter and get a comprehensive picture of the history of that concept. The only problem with this approach is that as you read the whole text you often get the feeling of déjà vu, as you find an entire paragraph that is repeated verbatim. I found the book to be enjoyable reading; it made me aware of the difficulties of introducing abstraction into mathematics and the role abstraction played in driving algebra in a new direction. It also made me reflect on how I, personally, would judge the relative significance of the various theories and contributing mathematicians.

Ueli Daepp, Associate Professor of Mathematics, Bucknell University

Ueli Daepp, reviewer, "A History of Abstract Algebra," *Loci* (April 2008)