When one picks up a book with a title like A First Graduate Course in Abstract Algebra, it is not clear what one will find inside the book. Different schools, and even different professors, seem to have vastly different ideas of what a course in Algebra should consist of, let alone what material belongs in the graduate, as opposed to the undergraduate curriculum. The new book of this title by W.J. Wickless starts at the very beginning of algebra. In fact, the first few chapters of the book cover the basics of group theory and ring theory and are mostly material that I would think of as undergraduate topics, albeit at a pace that would probably not be appropriate for undergraduates. While the book then discusses modules at a depth that I have never seen in an undergraduate textbook, the pace does not slow down as it does so. The book then quickly moves on to discuss vector spaces, fields, and Galois theory all at a level that I think of as more appropriate for an undergraduate text. It is only in the last several chapters that the book moves on to more advanced topics such as group extensions, noncommutative rings, and various structure theorems.
The book is well-written, and contains many nice exercises. However, there are lots of well-written introductory Algebra textbooks out there, and so it seems to me that a new book should bring something new to the table, and I am not convinced that Wickless does this. Mostly, I am not sure I understand who would find use for this book. It seems to me that many — if not most — graduate students would already have seen the majority of topics covered in the book, but I think that the pace of the book, as well as a lack of lots of good examples of such material, would make it very difficult for all but the best undergraduate students to use this book as an introduction to the material. I suppose the book would work well for students coming to mathematics from other disciplines at the graduate level, but I am not sure I think this target audience is large enough to justify another book on the already crowded bookshelves filled with introductory Algebra textbooks.
Darren Glass teaches at Gettysburg College.