When Skew Field Constructions appeared in 1977 in the London Mathematical Society Lecture Note Series, it was very much intended as a provisional text, to be replaced by a more definitive version. In the intervening years there have been some new developments, but most of the progress has been made in the simplification of the proofs of the main results. This has made it possible to include complete proofs in the present version, rather than to have to refer to the author's Free Rings and their Relations. An attempt has also been made to be more comprehensive, but we are without a doubt only at the beginning of the theory of skew fields, and one would hope that this book will offer help and encouragement to the prospective builders of such a theory. The genesis of the theory was described in the original preface (see the extract following this preface); below we briefly outline the subjects covered in the present book.

The first four chapters are to a large extent independent of each other and can be read in any order, referring back as necessary. Ch. 1 gives the general definitions and treats the Ore case as well as various necessary conditions for the embedding of rings in skew fields. From results in universal algebra it follows that necessary and sufficient conditions for such an embedding take the form of quasi-identities. Later, in Ch.4, we shall find the explicit form of these quasi-identities, and in Ch. 6 we shall see that this set must be infinite.