"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 a cornerstone of the theory of central simple algebras in the decades to come." (Uzi Vishne, Mathematical Reviews, October, 2016)

• 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.

"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 a cornerstone of the theory of central simple algebras in the decades to come." (Uzi Vishne, Mathematical Reviews, October, 2016)

• 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.