zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 12G05 Galois cohomology 12-02 Research monographs (field theory) 11E04 Quadratic forms over general fields 11E72 Galois cohomology of linear algebraic groups 12F12 Inverse Galois theory 16K20 Finite-dimensional division rings 16W10 Associative rings with involution, etc. 20G10 Cohomology theory of linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields 20J06 Cohomology of groups