Skew Fields: Theory of General Division Rings

Skew Fields: Theory of General Division Rings

by P. M. Cohn
     
 

Non-commutative fields (also called skew fields or division rings) have not been studied as thoroughly as their commutative counterparts and most accounts have hitherto been confined to division algebras, that is skew fields finite-dimensional over their centre. Based on the author's LMS lecture note volume Skew Field Constructions, the present work offers a… See more details below

Overview

Non-commutative fields (also called skew fields or division rings) have not been studied as thoroughly as their commutative counterparts and most accounts have hitherto been confined to division algebras, that is skew fields finite-dimensional over their centre. Based on the author's LMS lecture note volume Skew Field Constructions, the present work offers a comprehensive account of skew fields. The axiomatic foundation and a precise description of the embedding problem are followed by an account of algebraic and topological construction methods, in particular, the author's general embedding theory is presented with full proofs, leading to the construction of skew fields. The powerful coproduct theorems of G. M. Bergman are proved here as well as the properties of the matrix reduction functor, a useful but little-known construction providing a source of examples and counter-examples. The construction and basic properties of existentially closed skew fields are given, leading to an example of a model class with an infinite forcing companion which is not axiomatizable. The treatment of equations over skew fields has been simplified and extended by the use of matrix methods, and the beginnings of non-commutative algebraic geometry are presented, with a precise account of the problems that need to be overcome for a satisfactory theory. A separate chapter describes valuations and orderings on skew fields, with a construction applicable to free fields. Numerous exercises test the reader's understanding, presenting further aspects and open problems in concise form, and notes and comments at the ends of chapters provide historical background.

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Product Details

ISBN-13:
9780521062947
Publisher:
Cambridge University Press
Publication date:
05/15/2008
Series:
Encyclopedia of Mathematics and its Applications Series, #57
Pages:
520
Sales rank:
1,295,491
Product dimensions:
6.10(w) x 9.20(h) x 1.20(d)

Table of Contents

Preface
From the preface to Skew Field Constructions
Note to the reader
Prologue1
1Rings and their fields of fractions3
2Skew polynomial rings and power series rings47
3Finite skew field extensions and applications93
4Localization152
5Coproducts of fields202
6General skew fields278
7Rational relations and rational identities331
8Equations and singularities366
9Valuations and orderings on skew fields420
Standard notations473
List of special notations used throughout the text475
Bibliography and author index478
Subject index495

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