Value Functions on Simple Algebras, and Associated Graded Rings

This book explains Important advances in the theory of finite-dimensional division algebras have spurred the development of valuation theory in a noncommutative context since the latter part of the twentieth century. These advances include Amitsur's construction of noncrossed product algebras and Platonov's solution to the Tannaka-Artin problem. 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. This book is particularly timely because it approaches valuation theory 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 noncommutative valuation theory. Various constructions of division algebras are obtained as applications, such as noncrossed product algebras and indecomposable algebras. In addition, the use of valuation theory is showcased in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev). The intended audience consists of graduate students and research mathematicians.