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.

Review of the hardback:‘… the first book on this theme and will be the basis of any future development in this field.’

J. Schoissengeier Source: Monatshefte für Mathematik

Review of the hardback:‘While the material is quite technical, the book is very readable.’

Source: Mathematika

Review of the hardback:‘… an up-to-date account.’

Source: European Mathematical Society Newsletter

Review of the hardback:‘This is a tremendous piece of work, whose importance will grow for many years.’

Source: Bulletin of the London Mathematic Society