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Skew Fields
Theory of General Division Rings

Part of Encyclopedia of Mathematics and its Applications

  • Date Published: May 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521062947

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  • 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, is 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 theorem of G. M. Bergman is 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.

    • One of the world's most eminent algebraists
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    Reviews & endorsements

    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

    • Date Published: May 2008
    • format: Paperback
    • isbn: 9780521062947
    • length: 520 pages
    • dimensions: 229 x 152 x 15 mm
    • weight: 0.72kg
    • availability: Available
  • 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.

  • Author

    P. M. Cohn, University College London

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