Skew Fields: Theory of General Division Rings

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Algebraists have studied noncommutative fields (also called skew fields or division rings) less thoroughly than their commutative counterparts. Most existing accounts have been confined to division algebras, i.e. skew fields that are finite dimensional over their center. This work offers the first comprehensive account of skew fields. It is based on the author's LMS Lecture Note Volume "Skew Field Constructions". The axiomatic foundation and a precise description of the embedding problem precedes an account of algebraic and topological construction methods. The author presents his general embedding theory with full proofs, leading to the construction of skew fields. The author has simplified his treatment of equations over skew fields and has extended it by the use of matrix methods. 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. Notes and comments at the end of chapters provide historical background. The book will appeal to researchers in algebra, logic, and algebraic geometry, as well as graduate students in these fields.

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Editorial Reviews

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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. Rings and their fields of fractions; 2. Skew polynomial rings and power series rings; 3. Finite skew field extensions and applications; 4. Localization; 5. Coproducts of fields; 6. General skew fields; 7. Rational relations and rational identities; 8. Equations and singularities; 9. Valuations and orderings on skew fields; Standard notations; List of special notations used throughout the text; Bibliography and author index; Subject index.

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