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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.
Inhaltsverzeichnis
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; Bibliography; Index.
Über den Autor / die Autorin
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.
Zusammenfassung
The first comprehensive, modern introduction to a central field in modern algebra with connections to algebraic geometry, K-theory, and number theory. It proceeds from the basics to more advanced results, including the Merkurjev–Suslin theorem. It is ideal as a text for a graduate course and as a reference for researchers.
