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 audienceconsists of graduate students and research mathematicians.
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"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)