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.
Industry Reviews
`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.
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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.
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Mathematical Reviews
`This book is unique in its role of providing a self-contained and easily accessible introduction to the theory of orders and maximal orders in both the local and the global setting. Readers of the book will also find it valuable as a guide to many basic algebraic notions such as localizations, completions, the Jacobson radical, and Morita theory. The text is well-written and complete, and well-chosen exercises are offered at the end of each chapter. The
book is well-suited as a text for graduate courses in representation theory, and for many years has filled a gap in the literature in the area of orders and maximal orders.
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Professor Tsit-Yuen Lam, University of California at Berkeley.
