Overview
- Provides the first comprehensive treatment of valuation theory for finite-dimensional division algebras
- Showcases an innovative new approach to valuation theory via associated graded structures
- Includes various division algebra constructions with special features, like noncrossed products and indecomposable algebras
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (12 chapters)
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Front Matter
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Back Matter
Keywords
About this book
This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of non crossed products and Platonov's solution of the Tannaka-Artin problem.
This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased.
The intended audience consists of graduate students and research mathematicians.
Reviews
“The book gives a good idea of the present state of valuation theory on division algebras and its applications, and also, of a new approach to this field worked out with the active participation of the authors … . It can be very useful to graduate students and research mathematicians interested in central simple algebras, Brauer groups, valuation theory and other related areas. Each chapter contains examples and exercises … .” (Ivan D. Chipchakov, zbMATH 1357.16002, 2017)
“This monograph offers a detailed exposition of graded central simple algebras over graded fields and central simple algebras over valued fields, emphasizing strong connections between the two theories. … The two authors, who took major part in the development of this new theory, are masters of precise exposition. Their book will surely serve as acornerstone of the theory of central simple algebras in the decades to come.” (Uzi Vishne, Mathematical Reviews, October, 2016)
Authors and Affiliations
Bibliographic Information
Book Title: Value Functions on Simple Algebras, and Associated Graded Rings
Authors: Jean-Pierre Tignol, Adrian R. Wadsworth
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-319-16360-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2015
Hardcover ISBN: 978-3-319-16359-8Published: 16 April 2015
Softcover ISBN: 978-3-319-35587-0Published: 29 October 2016
eBook ISBN: 978-3-319-16360-4Published: 03 April 2015
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XV, 643
Topics: Associative Rings and Algebras, Field Theory and Polynomials