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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