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An introduction to Galois cohomology and its applications. (English) Zbl 1207.12003
London Mathematical Society Lecture Note Series 377. Cambridge: Cambridge University Press (ISBN 978-0-521-73866-8/pbk). xi, 315 p. $ 65.00; £ 40.00 (2010).
Galois cohomology is an important topic in algebra showing up in different places and in different guises, of great interest as a theory in its own right as well as because of its multifarious applications. There are various monographs that address certain aspects of Galois cohomology required in certain contexts, and there are of course the famous lecture notes [{\it J.-P. Serre}, Cohomologie Galoisienne. Lecture Notes in Mathematics. 5. Berlin etc.: Springer-Verlag (1973; Zbl 0259.12011)] (and its English translation [Galois cohomology. Springer Monographs in Mathematics. Berlin: Springer (2002; Zbl 1004.12003)]). But the present book seems to be the first comprehensive and largely self-contained introduction to (nonabelian) Galois cohomology and Galois descent techniques, building up the topic from first principles accessible even to advanced undergraduate or beginning graduate students with a decent grounding in algebra. Indeed, the first seven chapters grew out of a lecture course for graduate students given by the author at the University of Southampton. The book consists of two parts. Part I provides an introduction to Galois cohomology, Galois descent and some fairly accessible applications, making up about sixty percent of the book. The chapter headings give a good idea of how the theory is developed: Infinite Galois theory (ch. 1); Cohomology of profinite groups (ch. 2); Galois cohomology (ch. 3), this chapter contains an introduction to Galois descent; Galois cohomology of quadratic forms (ch. 4); Étale and Galois algebras (ch. 5); Group extensions, Galois embedding problems and Galois cohomology (ch. 6). This first part presents the theory in minute detail, making it also quite suitable for self-study. The second part deals with a range of applications, some classical, some quite recent, and they reflect to some extent the author’s personal taste and research interests. In the introduction, the author mentions other areas in which Galois cohomology has important applications and he provides the interested reader with the necessary references. This second part of the book is to some extent a survey on some applications of Galois cohomology, and it also serves as an introduction to other topics of intense current research activity in which Galois cohomology figures prominently. As such, the author does not claim this part to be self-contained, nor does he present each and every proof. But he does provide the reader with a rich source of background and material for further study in Galois cohomology and its applications. Again, the chapter headings give a good idea of the topics covered: Galois embedding problems and trace form (ch. 7); Galois cohomology of central simple algebras (ch. 8); Digression: a geometric interpretation of $H^1(-,G)$ (ch. 9); Galois cohomology and Noether’s problem (ch. 10); The rationality problem for adjoint algebraic groups (ch. 11); Essential dimension of functors (ch. 12). As a nice added feature, each chapter is complemented by a section with exercises. This book is a very welcome addition to the literature for people doing research in Galois cohomology or using it as a tool in their research or in lecture courses.
MSC:
12G05Galois cohomology
12-02Research monographs (field theory)
11E04Quadratic forms over general fields
11E72Galois cohomology of linear algebraic groups
12F12Inverse Galois theory
16K20Finite-dimensional division rings
16W10Associative rings with involution, etc.
20G10Cohomology theory of linear algebraic groups
20G15Linear algebraic groups over arbitrary fields
20J06Cohomology of groups