Algebra II: Fields with Structure, Algebras and Advanced Topics is the second of two volumes that form a complete algebra course, including both undergraduate and graduate topics. (See also volume one, which deals with Fields and Galois Theory. This review discusses both volumes.) The only background assumed is linear algebra. Frequent references are made to the author's book on the subject, which unfortunately has not been translated into English. (An appendix listing the exact results from linear algebra that are required would have made it easier for the reader to find proofs if necessary.)
The first volume uses Galois theory as a guiding line that leads the reader to excursions into arithmetic, groups, rings, and of course fields. It starts with the classical Greek problem of constructibility with ruler and compass and unfolds Galois theory from there, ending with Drinfeld's proof of the transcendence of π.
In its use of field theory as a guiding theme, Lorenz's book reminded me of Hadlock's Field Theory and Its Classical Problems. However, Hadlock's engaging chatty style contrasts with Lorenz's efficient, but lively, rendering of the traditional definition-theorem-proof commonly attributed to Landau. (Lorenz departs from tradition by labeling propositions and lemmas as F1, F2, ...; F for Feststellung.) Lorenz's exposition is not interrupted by exercises, which are collected at the end of each volume.
The second volume focuses on fields with structure and algebras. It covers real fields, absolute values and valuations, sum of squares and Hilbert's 17th problem, p-adic completions, local fields, and much more, including an interesting result by Tsen on the existence of solutions of polynomial systems that is elementary but not well known. An important theme in this volume is the theory of central-simple algebras, leading to Brauer groups, crossed products, and a cohomological proof of local class field theory. This material is much more difficult than the material in the first volume and does not appear in general books at this level (a partial exception is Jacobson's Basic Algebra), but Lorenz manages to keep the exposition as smooth as in the first volume.
There is no group theory per se, no commutative algebra, etc. Only the results needed for the main threads of the book are discussed. Nevertheless, the choice of topics and their organization are excellent and provide a unifying view of most of algebra. In all, Lorenz's book is a wonderful reference for both teachers and researches, and can be used with much profit for independent study by hard-working students.
Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.
Foreword.- Ordered fields and real fields.- Hilbert's seventeenth problem and the real nullstellensatz.- Orders and quadratic forms.- Absolute values on fields.- Residue class degree and ramification index.- Local fields.- Witt vectors.- The tsen rank of a field.- Fundamentals of modules.- Wedderburn theory.- Crossed products.- The brauer group of a local field.- Local class field theory.- Semisimple representations of finite groups.- The schur group of a field.- Appendix: problems and remarks.- Index.