Almost a year ago (May of 2007, in fact) I had the pleasure of reviewing, and recommending, Basic Algebra, by Anthony W. Knapp, which the author offered as the first part of an extensive and far-reaching university course in algebra, taking the student from the immediately-after-calculus stage to beyond what is needed for the algebra qualifying examinations at a good graduate school. The present book, aptly titled *Advanced Algebra*, constitutes the concluding part of the course, starting with a chapter on number theory and ending with a chapter on algebraic geometry. With this supplementary offering, Knapp succeeds beautifully in achieving his stated minimal purpose for this pair of texts, namely, to prepare an aspiring mathematician to listen intelligently to a plenary convention lecture on an algebraic topic.

The material in *Advanced Algebra* is well-chosen and evinces a classical sensibility on the author’s part. For example, the initial arithmetical motivation features quadratic reciprocity, quadratic forms, and primes in arithmetic progressions (including Dirichlet’s proof: the requisite Fourier analysis was taken care of in *Basic Algebra*). Subsequently such exciting subjects as Wedderburn theory, the Brauer group (with *Satz* 90 and some group cohomology included), and homological algebra (up to and including Ext, Tor, etc.) are covered, soon to be followed by more number theory. Indeed, Knapp presents three real jewels of arithmetic right in the middle of the book: Dedekind’s discriminant theorem (*p* ramifies in K/**Q** iff *p* is a divisor of Disc(K)), Dirichlet’s unit theorem, and the theorem on the finiteness of the class number. As far as I am concerned, it does not get any better than this.

Knapp uses these arithmetical expositions as a motivation for the final set of themes developed in *Abstract Algebra*, entailing a thorough presentation of adèles and idèles, infinite field extensions, and three chapters on algebraic geometry. The topics covered here should move the reader to further study from e.g. Weil’s (anything but) Basic Number Theory, Serre’s *Cohomologie Galoisienne* (now available in English as *Galois Cohomology*, of course), and Griffiths and Harris’ *Principles of Algebraic Geometry*, respectively. (If I may be allowed a pair of digressions, I would actually opt for Neukirch’s remarkably compact *Class Field Theory* instead of Serre’s book, although Knapp only gives a footnote’s coverage (p. 265) to *Klassenkörpertheorie*; and, since he is equally terse as regards sheaf cohomology (p. 218) — except for the very last section of the book (on schemes) — I do not think Hartshorne’s *Algebraic Geometry* is indicated. On the other hand, can, or should, any one interested in algebraic geometry neglect Hartshorne?)

In any case, *Advanced Algebra* is a wonderfully useful and well-written book, characterized by clear and “user-friendly” treatments of many important algebraic topics. For instance, on p. 205, in his discussion of homological algebra, Knapp presents a welcome synopsis of the mechanics of derived functors in the format of a six-column table: we read off, for instance, that a right exact covariant functor requires a projective resolution (in a category of R-modules, say) to provide its homology by means of left derived functors, and the functor M ⊗_{R} (_) is given as an exemplar.

Another illustration of Knapp’s desire to situate his discussions in an accessible setting is his introduction to the chapter, “The Number Theory of Algebraic Curves” (Ch. IX, p. 520 ff.), consisting in a terrific treatment of the subject’s historical roots, followed by two brief sections, titled, respectively, “What to expect from the study,” and “Key observation to be used during the study.” This pedagogical sensitivity is present throughout *Advanced Algebra* (as is the case for *Basic Algebra*, also): Knapp’s chapter abstracts are another good indication of this commendable feature.

Finally, *Advanced Algebra* contains a thorough coverage of Gröbner bases, as any contemporary treatment of algebra at this level should, and continues the trend set in *Basic Algebra* of providing good, meaningful (and plentiful) exercises. I highly recommend this wonderful book.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.