Skew Fields

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.