This is the latest entry in a series of books by Jeremy Gray about the history of mathematics in (mostly) the 19th century. I say “mostly”, of course, because abstract algebra did not spring into existence in the 19th century but instead developed gradually over the years, and the book traces many of these 17th and 18th century developments as well as the refinements that took place in the 1800s and early 1900s. It does for abstract algebra what the author’s earlier

*Worlds Out of Nothing* and

*The Real and the Complex* did for geometry and analysis, respectively. Like these other two texts, this is a nice book to have around; it reflects careful scholarship and is filled with interesting material.

This book, like its predecessors, has 30 chapters (well, *Worlds* has 31) and is based on a series of thirty lectures given by the author at the University of Warwick. Some of these chapters reflect the book’s classroom origins. Several, for example, consist of “revisions” where the author summarizes what has been done and offers proposed essay topics. There are also helpful comments aimed directly at students, intending to give them some sense of what a historian does and how he or she does it.

Several historical threads wind their way through this book. One is how the study of numbers eventually led to commutative algebra and algebraic number theory. Accordingly, the text begins with a number of topics in number theory that were influential in the development of algebra, including quadratic forms and early attempts to prove Fermat’s Last Theorem. Gauss’ major work *Disquisitiones Arithmeticae* is discussed in some detail over the course of several chapters. Another major thread concerns solutions of polynomial equations, which takes us from the work of Cardano and Ferrari to Lagrange, Abel, Ruffini and ultimately to Galois theory and its aftermath. Several other chapters discuss the “modernization” of algebra in the late 19th and early 20th centuries, focusing on the significance of people like Noether, Weber and van der Waerden. The evolution of the concept of a group is also traced.

No book of this length can possibly cover *every* important topic in the history of abstract algebra, of course. This one omits any real discussion of the history of linear algebra, and, even among topics that more obviously fall within the title “abstract algebra”, there are some notable omissions: quaternions, for example, are barely mentioned. (The algebra considered in this book is mostly commutative.) Lie algebras, first introduced in the late 1800s, are also not covered here.

Though this book is based on lecture notes from a course (in the UK), I have some doubts as to its suitability as a text for an undergraduate course here in the United States. For one thing, there are relatively few exercises — about a third of the chapters have a section marked “exercises”, containing about three or four of them, and there are also some exercises embedded in the text itself.

In addition to the relative paucity of exercises, there are other reasons why this book would not likely fare well as a text for a course in the history of mathematics or history of algebra. It is quite demanding, for example, and covers some substantial mathematics (including a proof of the Law of Quadratic Reciprocity). I’m not sure that many students in a history of mathematics course will have the wherewithal to understand the level of mathematics that is presented here.

I also have a few other nits to pick. The book’s Index, for one thing, is woefully inadequate. Although titled “Index”, it is essentially only a “name index”; the few actual subjects that appear in it are those that are strongly associated to a particular name, such as “Galois field” or “Galois group”. The words “group”, “ring”, “field” and “ideal”, for example, do not appear as index entries. (The name “Gauss” has about 45 page entries, so have fun if you want to use the Index to read about what he had to say about quadratic reciprocity.) Even some names are missing from the index; Riemann’s name, for example, does not appear in it, even though he is mentioned in the text.

Another problem concerns typos, misprints and printing errors. There are probably not as many of these as there were in Gray’s earlier book on analysis, but there are still enough to be noticeable. For one thing, in my print copy of the book, the index ends with the letter “S”, so that names like Weber and van der Waerden do not appear at all. (My PDF copy of the book, downloaded from SpringerLink, does have an index that extends to “Z”, which again perhaps reflects my rapidly growing belief that Springer seems more interested in the electronic market than the print one.)

In addition to this missing index page, there are other typos, ranging from trivial things like the occasional missing word (“if we fail prove”; “as if it was square of a prime number”) to more problematic issues. Felix Klein’s famous address on geometry, for example, is referred to in the Introduction as the “Erlangen programme” but in Appendix E as the “Erlanger Programm”. In addition, the description of irreducible elements in the ring of Gaussian integers on page 203 is incorrect: Gray neglects to mention that an associate of any element in the list is also irreducible. Some typos are unintentionally amusing, as, for example, these two sentences on page 262: “It is trivial that a primary ideal is prime. The simplest example of a primary ideal that is not prime is…”.

Notwithstanding these issues, there is much to like about this book. It is quite detailed, contains a lot of information, is meticulously researched, and has an extensive bibliography. Anyone interested in the history of mathematics, or abstract algebra, will want to make the acquaintance of this book. I gather from the Preface in

*The Real and the Complex* that a fourth historical volume (on the history of differential equations) was, at least as of that time, planned; I look forward to seeing it.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.