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Advanced algebra. Along with a companion volume ‘Basic algebra’. (English) Zbl 1133.00001
Cornerstones. Basel: Birkhäuser (ISBN 978-0-8176-4522-9/hbk; 978-0-8176-4533-5/set). xxiv, 730 p. EUR 54.90/net, SFR 95.00, $69.95, £ 42.00; EUR 69.90, SFR 125.00,$ 89.95, £ 54.00/set (2007).

This textbook is the second volume of Anthony W. Knapp’s comprehensive introduction to the fundamental concepts and tools in modern abstract algebra. Together with its foregoing companion volume “Basic Algebra” [Basel: Birkhäuser (2006; Zbl 1106.00001)], which was published in the autumn of 2006, the current book is to provide a global view of the subject, thereby particularly emphasizing both its various applications and its ubiquitous role in contemporary mathematics. As the author already pointed out in the preface to the first volume, his leading idea was to give a systematic account of what a budding mathematician needs to know about the principles of modern algebra in order to communicate well with colleagues in all branches of mathematics and related sciences.

This rewarding program was masterly begun in the companion volume “Basic Algebra”, where the fundamentals of linear algebra, multilinear algebra, group theory, commutative algebra, field theory, Galois theory, and module theory over noncommutative rings were profoundly developed. As for the author’s particular expository guidelines, exemplary didactic principles, and his notorious brilliant style of lucid mathematical writing, we may refer to the review of the first volume (the author, loc. cit.), as these outstanding features, which also characterize the second volume “Advanced Algebra” under review to full extent, have been depicted and appraised there at great length.

In general the present volume assumes knowledge of most of the content of its forerunner “Basic Algebra”, either from that book itself or from some comparable source. The more advanced topics treated in the current book mainly point toward algebraic number theory and algebraic geometry, with emphasis on aspects of these subjects that impact fields of mathematics other than algebra. In this vein, the predominant theme is the fundamental and fascinating interrelation between number theory and geometry, where this aspect constantly recurs throughout the book on different levels.

As to the precise contents, the volume under review consists of ten chapters, each of which is subdivided into several sections.

Chapter 1 is titled “Transition to Modern Number Theory” and discusses three classical results of Gauss and Dirichlet that were milestones in the transition from the early number theory of Fermat, Euler, and Lagrange to the algebraic number theory of Kummer, Dedekind, Kronecker, Hermite, and Eisenstein in the second half of the 19th century. Concretely, after an introductory section on the historical background of this transition process, this chapter establishes Gauss’s Law of Quadratic-Reciprocity, the theory of binary quadratic forms, and Dirichlet’s Theorem on primes in arithmetic progressions, including the basics of quadratic number fields, their units, Dirichlet series, and Euler products.

Chapter 2 is devoted to the theory of finite-dimensional associative algebras, division algebras, and closely related classes of rings. The material covered here is part of what is known as Wedderburn-Artin ring theory, and it comprises the following topics: semisimple rings and Wedderburn’s Theorem, rings with chain conditions and Artin’s Theorem for simple rings, the Wedderburn-Artin radical and Wedderburn’s Main Theorem, semisimplicity and tensor products, the Skolem-Noether Theorem, the Double Centralizer Theorem, Wedderburn’s Theorem about finite division rings, and Frobenius’s Theorem about real division algebras.

The further study of associative algebras is the subject of Chapter 3, with special emphasis on the Brauer group of a field as a fundamental tool for classifying noncommutative division rings. Group cohomology, the interpretation of the Brauer group in this cohomological context, crossed products, and Hilbert’s Theorem 90 tie associative algebras to algebraic number theory, and this link is thoroughly explained in the course of this third chapter.

The rudiments of the subject of general homological algebra are subsequently developed in Chapter 4. Using the digression on group cohomology in the previous chapter as motivation, the author treats the basics of homology theory in the context of “good” categories of modules over a ring, with an extension of the discussion to homological algebra in general Abelian categories in the final section of this chapter. The standard topics in homological algebra, including complexes and additive functors, long exact homology sequences, injective and projective objects, derived functors and their long exact sequences, the functors Ext and Tor and the algebra of Abelian categories, are taken up here to a remarkable extent. Having the methods of cohomology available at this point of the present book means that the reader is well prepared for its use in both algebraic number theory and algebraic geometry, which are the main themes in the remaining six chapters.

Chapter 5 deals with three important theorems in the theory of algebraic number fields and their rings of integers, namely with the Dedekind Discriminant Theorem, the Dirichlet Unit Theorem, and the theorem on the finiteness of the class number of a number field. The direct approach adopted here is generalized in the subsequent Chapter 6 where the reinterpretation in the modern conceptual framework of the theory of “adèles” and “idèles” is provided. In fact, this chapter develops some of the advanced tools for a more penetrating study of algebraic number theory, among which the reader encounters $p$-adic numbers, discrete valuations, absolute values, completions of fields, Hensel’s Lemma, ramification indices and residue class degrees, differents and discriminants, global and local fields, Artin’s product formula, the ring of adèles, and the idèle-class group of a global field.

Chapter 7 provides some algebraic background material for the later study of fundamental questions in algebraic geometry. This includes the Hilbert Nullstellensatz, the transcendence degree of an infinite field extension, separable and purely inseparable field extensions, the Krull dimension of a ring, regular and singular points of affine varieties, infinite Galois groups, and profinite groups.

In the following, an introduction to the foundations of algebraic geometry is given from three different points of view.

Chapter 8 basically approaches algebraic geometry in its purely algebraic setting, that is, as a framework to study solutions of simultaneous solutions of polynomial equations in several variables by means of ideal-theoretic and module-theoretic methods. This is done in the light of the theory of projective plane curves and their intersection multiplicities, in the first part, and of the computational approach via Gröbner bases in the sequel. In the final section, these two approaches are combined to derive the crucial Elimination Theorem in its full generality.

Chapter 9 treats the subject of algebraic curves as the classical outgrowth of the complex analysis of compact Riemann surfaces, on the one hand, and of its arithmetic roots on the other. The leading theme is here the strong analogy between one-dimensional function fields and algebraic number fields. While the first sections define divisors and the genus of an arithmetic curve (or of a compact Riemann surface, respectively), the last two sections give a detailed proof of the Riemann-Roch Theorem for such curves and illustrate some of its important applications. The fundamental tool for the author’s approach is the theory of discrete valuations (as developed in Chapter 6), through which the parallel between the arithmetic of number fields and the geometry of curves becomes strikingly evident.

The final Chapter 10 turns from curves to general affine and projective algebraic varieties. The topics touched upon here include affine varieties, the concept of geometric (Noether) dimension, projective varieties, rational functions and regular functions, morphisms of affine and projective varieties, rational maps between them, Zariski’s Theorem about smooth points, classification questions about irreducible curves, affine varieties defined by monomial ideals, Hilbert polynomials, and intersection properties of projective varieties derived from Hilbert polynomials. The book ends with an informal outlook to algebraic schemes (à la A. Grothendieck), together with a number of hints for further reading in this much more advanced direction, which the keen reader of the current text should be well prepared for, after having sucessfully mastered the study of it.

Each chapter comes with its own detailed introduction, its own historical remarks, and its own collection of carefully selected problems. These problems are intended to play an important role within the entire text, because many of them provide additional, further-going topics enhancing the core material of the book. However, almost all problems are solved in the extra section of hints at the end of the book, which both helps the reader control her or his understanding of the wealth of fundamental material and get acquainted with a large number of additional concepts, methods, theorems, examples, and applications. As in the first volume of this comprehensive textbook, there is a detailed guide for the reader how to use this book, a chart of the main lines of dependence among the single chapters, a list of some items of notation and terminology from the first volume “Basic Algebra”, an index of notation for the present volume, and a rich bibliography referring to related textbooks on the subjects discussed in the present book.

All together, this is another outstanding textbook written by the renowned and versatile mathematical researcher, teacher, and author Anthony W. Knapp that reflects his spirit, his devotion to mathematics, and his rich experiences in expository writing at best.

##### MSC:
 00A05 General mathematics 11-01 Textbooks (number theory) 14-01 Textbooks (algebraic geometry) 16-01 Textbooks (associative rings and algebras) 18-01 Textbooks (category theory) 12-01 Textbooks (field theory)