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Research and Exposition in Mathematics -- Volume 2

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J. Dauns

A Concrete Approach to Division Rings

438 p., soft cover, ISBN 3-88538-202-4, EUR 12.00, 1982

This is the first book which treats all types of division rings. Traditionally, the theory of division rings has been based on heavy algebra, thus restricting its accessibility to specialists. In contrast, the first objective of this book is to develop the important basic facts quickly in as straightforward a manner as possible.

In the classical references on division rings the plenitude of theorems contrasts sharply with the paucity of examples illustrating them. Another aim of this book thus is to develop the subject via examples. Many such concrete examples had to be invented and constructed, and are presented here for the first time.

The book begins by considering the classical quaternions, then generalized quaternions, and then cyclic algebras. The author next defines arbitrary crossed products, and finally the universal division algebras. Amitsur's noncrossed product proof is outlined. Along the way, the book covers part of the elementary theory of division algebras and the Brauer group. The second half of this book treats some constructions of infinite-dimensional division algebras. Central to the discussion are twisted polynomial rings. Power series rings and a class of finite extensions termed "pseudolinear" are also covered.

Most chapters of the book can be read independently and do not depend on the rest of the book.