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Skew Fields: Theory of General Division Rings

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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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Editorial Reviews

From the Publisher
Review of the hardback: '… the first book on this theme and will be the basis of any future development in this field.' J. Schoissengeier, Monatshefte für Mathematik

Review of the hardback: 'While the material is quite technical, the book is very readable.' Mathematika

Review of the hardback: '… an up-to-date account.' European Mathematical Society Newsletter

Review of the hardback: 'This is a tremendous piece of work, whose importance will grow for many years.' Bulletin of the London Mathematic Society

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Product Details

Table of Contents

Preface
From the preface to Skew Field Constructions
Note to the reader
Prologue 1
1 Rings and their fields of fractions 3
2 Skew polynomial rings and power series rings 47
3 Finite skew field extensions and applications 93
4 Localization 152
5 Coproducts of fields 202
6 General skew fields 278
7 Rational relations and rational identities 331
8 Equations and singularities 366
9 Valuations and orderings on skew fields 420
Standard notations 473
List of special notations used throughout the text 475
Bibliography and author index 478
Subject index 495
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