This book is the first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Starting from the basics, it reaches such advanced results as the Merkurjev-Suslin theorem. This theorem is both the 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, but no homological algebra, the book covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi-Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. The last chapter rounds off the theory by presenting the results in positive characteristic, including the theorem of Bloch-Gabber-Kato. The book is suitable as a textbook for graduate students and as a reference for researchers working in algebra, algebraic geometry or K-theory.

• Modern, comprehensive introduction assuming only a solid background in algebra, but no homological algebra; necessary results from algebraic geometry are summarized in an appendix • Accessible proof of the Merkurjev-Suslin theorem • First textbook treatment of characteristic p methods, including the Jacobson-Cartier and Bloch-Gabber-Kato theorems

### Contents

1. Quaternion algebras; 2. Central simple algebras and Galois descent; 3. Techniques from group cohomology; 4. The cohomological Brauer group; 5. Severi-Brauer varieties; 6. Residue maps; 7. Milnor K-theory; 8. The Merkurjev-Suslin theorem; 9. Symbols in positive characteristic; Appendix: A breviary of algebraic geometry; References; Index.

### Review

'The presentation of material is reader-friendly, arguments are clear and concise, exercises at the end of every chapter are original and stimulating, the appendix containing some basic notions from algebra and algebraic geometry is helpful. To sum up, the book under review can be strongly recommended to everyone interested in the topic.' Zentralblatt MATH