Research and Exposition in Mathematics -- Volume 2
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.