Overview

This is a reissue of a classic text, which includes the author's own corrections and provides a very accessible, self contained introduction to the classical theory of orders and maximal orders over a Dedekind ring. It starts with a long chapter that provides the algebraic prerequisites for this theory, covering basic material on Dedekind domains, localizations and completions, as well as semisimple rings and separable algebras. This is followed by an introduction to the basic tools in studying orders, such as reduced norms and traces, discriminants, and localization of orders. The theory of maximal orders is then developed in the local case, first in a complete setting, and then over any discrete valuation ring. This paves the way to a chapter on the ideal theory in global maximal orders, with detailed expositions on ideal classes, the Jordan-Zassenhaus Theorem, and genera. This is followed by a chapter on Brauer groups and crossed product algebras, where Hasse's theory of cyclic algebras over local fields is presented in a clear and self-contained fashion. Assuming a couple of facts from class field theory, the book goes on to present the theory of simple algebras over global fields, covering in particular Eichler's Theorem on the ideal classes in a maximal order, as well as various results on the KO group and Picard group of orders. The rest of the book is devoted to a discussion of non-maximal orders, with particular emphasis on hereditary orders and group rings. The ideas collected in this book have found important applications in the smooth representation theory of reductive p-adic groups. This text provides a useful introduction to this wide range of topics. It is written at a level suitable for beginning postgraduate students, is highly suited to class teaching and provides a wealth of exercises.

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9780198526735

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Reviews

Reiner's book provides an excellent introduction for students and serves as an indispensible reference for researchers. Zentralblatt MATH Reiner's book gives by far the most extensive and most readable account available of the classical theory of maximal orders. The book has been written with great care, and is a pleasure to read. Unlike many books at such an advanced level, it contains many interesting exercises, with hints where appropriate. It is essential to the library of every working algebraist. Bulletin of the American Mathematical Society The book certainly fills a gap in the mathematical literature, since no modern text-book on maximal orders has been available. The author has succeeded very well in giving a clear and easily accessible presentation of the subject. Mathematical Reviews

Reiner's book provides an excellent introduction for students and serves as an indispensible reference for researchers. Zentralblatt MATH Reiner's book gives by far the most extensive and most readable account available of the classical theory of maximal orders. The book has been written with great care, and is a pleasure to read. Unlike many books at such an advanced level, it contains many interesting exercises, with hints where appropriate. It is essential to the library of every working algebraist. Bulletin of the American Mathematical Society The book certainly fills a gap in the mathematical literature, since no modern text-book on maximal orders has been available. The author has succeeded very well in giving a clear and easily accessible presentation of the subject. Mathematical Reviews

Author Information

Professor Irving Reiner (1924-1986), was one of the world's leading experts in representation theory. During his life he published more than 80 research papers, four books (including the original issue of Maximal Orders published by Academic Press in 1975) and many research survey articles on topics related to those contained in this text. In 1962 he was the John Simon Guggenheim Fellow and a former editor of the Illinois Journal of Mathematics and a long-time member of the American Mathematical Society.

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