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Full Description
The book is written in three parts. Part I consists of preparatory work on algebras, needed in Parts II and III. This material is presented in a classical, though unusual, way. Part II consists of a modern description of the theory of Brauer groups over fields (from as elementary a point of view as possible). Part III covers some new developments in the theory which, until now, have not been available except in journals. The principal topic discussed in this section is reduced K,-theory. This book will be of interest to graduate students in pure mathematics and to professional mathematicians.
Contents
Preface; Conventions on terminology; Part I. Skew Fields and Simple Rings: 1. Some ad hoc results on skew fields; 2. Rings of matrices over skew fields; 3. Simple rings and Wedderburn's main theorem; 4. A short cut to tensor products; 5. Tensor products and algebras; 6. Tensor products and Galois theory; 7. Skolem-Noether theorem and Centralizer theorem; 8. The corestriction of algebras; Part II. Skew Fields and Brauer Groups: 9. Brauer groups over fields; 10. Cyclic algebras; 11. Power norm residue algebras; 12. Brauer groups and Galois cohomology; 13. The formalism of crossed products; 14. Quaternion algebras; 15. p-Algebras; 16. Skew fields with involution; 17. Brauer groups and K2-theory of fields; 18. A survey of some further results; Part III. Reduced K1-Theory of Skew Fields: 19. The Bruhat normal form; 20. The Dieudonné determinant; 21. The structure of SLn (D) for n ≥ 2; 22. Reduced norms and traces; 23. The reduced Whitehead group SK1 (D) and Wang's theorem; 24. SK1 (D) ≠ 1 for suitable D; Remarks on USK1 (D,I); Bibliography; Thesaurus; Index.
