Skew Fields

P. K. Draxl

doi:10.1017/CBO9780511661907

Skew Fields

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.

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Skew Fields

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Conventions on Terminology
pp ix-x

Part I - Skew Fields and Simple Rings
pp 1-2

1 - Some ad hoc Results on Skew Fields
pp 3-7

2 - Rings of Matrices over Skew Fields
pp 8-12

3 - Simple Rings and Wedderburn's Main Theorem
pp 13-17

4 - A Short Cut to Tensor Products
pp 18-24

5 - Tensor Products and Algebras
pp 25-32

6 - Tensor Products and Galois Theory
pp 33-38

7 - Skolem–Noether Theorem and Centralizer Theorem
pp 39-49

8 - The Corestriction of Algebras
pp 50-56

Part II - Skew Fields and Brauer Groups
pp 57-58

9 - Brauer Groups over Fields
pp 59-70

10 - Cyclic Algebras
pp 71-76

11 - Power Norm Residue Algebras
pp 77-91

12 - Brauer Groups and Galois Cohomology
pp 92-96

13 - The Formalism of Crossed Products
pp 97-102

14 - Quaternion Algebras
pp 103-105

15 - p-Algebras
pp 106-111

16 - Skew Fields with Involution
pp 112-118

17 - Brauer Groups and K2-Theory of Fields
pp 119-121

18 - A Survey of some further Results
pp 122-124

Part III - Reduced K1-Theory of Skew Fields
pp 125-126

19 - The Bruhat Normal Form
pp 127-132

20 - The Dieudonné Determinant
pp 133-139

21 - The Structure of SLn(D) for n ≥ 2
pp 140-142

22 - Reduced Norms and Traces
pp 143-154

23 - The Reduced Whitehead Group SK1(D) and Wang's Theorem
pp 155-166

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