Value Functions on Simple Algebras, and Associated Graded Rings

by Jean-Pierre Tignol, Adrian R. Wadsworth

Overview

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.

Product Details

ISBN-13:	9783319355870
Publisher:	Springer International Publishing
Publication date:	04/16/2016
Series:	Springer Monographs in Mathematics
Edition description:	Softcover reprint of the original 1st ed. 2015
Pages:	643
Product dimensions:	6.10(w) x 9.25(h) x 0.05(d)

Table of Contents

Valuations on Division Rings.- Graded Algebra.- Value Functions.- Existence and Fundamental Properties of Gauges.- Graded and Valued Field Extensions.- Brauer Groups.- Total Ramifcation.- Division Algebras over Henselian Fields.- Subfields and Splitting Fields of Division Algebras.- Indecomposable Division Algebras.- Computation of SK1(D).- The Essential Dimension of Central Simple Algebras.

Editorial 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)

From the Publisher