Cornerstones. Basel: Birkhäuser (ISBN 0-8176-3248-4/hbk; 0-8176-4533-0/set). xxii, 717 p. EUR 58.00, SFR 92.00; EUR 69.90/net, SFR 125.00, $ 89.95, £ 54.00/set (2006).

Anthony W. Knapp is a renowned mathematical researcher, teacher, and textbook author whose excellent monographs “Elliptic Curves” (Mathematical Notes 40, Princeton Univ. Press, 1992;

Zbl 0804.14013) and “Lie Groups Beyond an Introduction” (Progress in Mathematics 140, Birkhäuser Verlag, 1996;

Zbl 0862.22006), among others, have “become standard references in their respective fields. Now, as a Professor Emeritus, this highly experienced teacher has undertaken the gratifying project to publish a series of foundational textbooks covering the essentials of real analysis and algebra, thereby sharing both his rich experiences and his outstanding expertise with the mathematical community as a whole. In the new series “Cornerstones” of Birkhäuser Verlag, which is to be devoted to publishing fundamental textbooks in mathematics, his two companion volumes “basic real analysis” and “advanced real analysis” have already appeared in 2005 (see

Zbl 1095.26001 and

Zbl 1095.26002), and his two other companion volumes” “basic algebra” and “advanced algebra” are about to follow right away. The book under review, is the first volume of Anthony W. Knapp’s announced algebra text. Together with its companion volume “advanced algebra”, which is to appear by September 2007 (see

Zbl 1133.00001), it is designed to systematically develop those fundamental concepts and methods of modern abstract algebra that are indispensable for every active mathematician in our days. The author’s main goal is to provide a global view of fundamental algebra, its various applications, and its ubiquitous role in contemporary mathematics. As the author points out in the preface to his book, the leading idea is to explain what the young mathematician needs to know about modern abstract algebra in order to communicate well with colleagues in all branches of mathematics and related sciences. In this vein, the primary audience of the two companion textbooks is students who encounter the material for the first time and who are planning a career in which advanced mathematics will play a predominant role. As for the contents of the present volume “basic algebra”, the book consists of ten chapters and six appendices. Much of the material covered here seems to correspond to standard course work, but most of the chapters include many further topics beyond that. This makes the entire text very flexible with regard to teaching or self-study likewise, and the reader has various choices to arrange her or his studies according to the respective needs. The precise contents of the present volume are as follows: Chapter 1 provides some preliminaries about integers, polynomials, and matrices, thereby emphasizing the according unique factorization properties and matrix operations, respectively. Chapter 2 develops the basics of linear algebra over the standard ground fields $\bbfQ$, $\bbfR$, and $\bbfC$. Chapter 3 introduces inner-product spaces up to the spectral theorem. Chapter 4 turns to basic group theory and group actions, with a first treatment of categories and functors at the end. Chapter 5 discusses the endomorphisms of finite-dimensional vector spaces, culminating in the primary decomposition theorem, the Jordan normal form, and their applications. Chapter 6 treats bilinear forms and, more generally, the basic concepts from multilinear algebra, including tensor algebra, symmetric algebra, and exterior algebra. Chapter 7 offers more advanced group theory, with the focus on free groups, free products, and group representations. Additional topics in this chapter are Burnside’s theorem and the elements of group extensions. Chapter 8 is devoted to the fundamental concepts and methods of commutative algebra. This chapter explains rings and modules, prime and maximal ideals, unique factorization domains, the structure of finitely generated modules over a principal ideal domain, noetherian rings, integral dependence, localization and local rings, and the structure of Dedekind domains. Chapter 9 develops the general theory of field extensions and then goes on to study Galois groups and their various applications to algebraic equations and geometrical constructions. The final Chapter 10 deals with the structure theory of modules over noncommutative rings, with special emphasis put on finiteness conditions and functorial properties of Hom and tensor products for such modules. The set of six appendices, included for the convenience of the reader, compile some basic facts from set theory and number domains that are used throughout the text. As a special feature, the text comes with numerous concrete examples and hundreds of selected exercises, and a separate 90-page section at the end of the book provides detailed hints or even complete solutions for most of the working problems. Each chapter comes with its own guiding abstract, and general instructions for using this textbook are given right after the thorough preface. The key features of this outstanding algebra text can be summarized as follows: (1) The basics of algebra are taught by stressing their close interrelations, and they are built on each other continually. (2) The three prominent themes, namely the analogy between integers and polynomials, the interplay between linear algebra and group theory, and the interaction between number theory and geometry, recur and blend together frequently. (3) Applications of abstract algebra to different areas of mathematics, natural sciences, and engineering appear repeatedly, often so in sequences of guided working problems (with solutions at the end of the book). (4) The author presents the subject matter in a forward-looking way by taking its natural development into account and proceeding from the particular to the general, thereby setting high value on the didactic aspects of teaching modern abstract algebra. (5) Computational methods are emphasized throughout the entire text in order to combine the abstract with the concrete tools. Finally, the author’s notorious masterly style of writing, which stands out by its high degree of clarity, elegance, refinement, and accuracy, also rules over this newest textbook of his, which is very likely to become one of the great standard texts in algebra for generations. The author has already given an outlook to the contents of the forthcoming companion volume “advanced algebra”, which will treat topics from modern number theory, Wedderburn-Artin ring theory, Brauer groups, homological algebra, infinite field extensions, basic algebraic geometry, and the arithmetic of algebraic curves. The mathematical community should be looking forward to the appearance of this second volume in the near future!