The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics. The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results. Philippe Gille is a Research Director for Centre National de la Recherche Scientifique at Institut Camille Jordan, Lyon. He has written numerous research papers on linear algebraic groups and related structures. Tamás Szamuely is a Research Advisor at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest and a Professor at the Central European University, Hungary. He is the author of Galois Groups and Fundamental Groups (Cambridge, 2009), also published in the Cambridge Studies in Advanced Mathematics series, as well as numerous research papers.