Prologue

Summary

O glücklich, wer noch hoffen kann,

Aus diesem Meer des Irrtums aufzutauchen!

Was man nicht weiss, das eben brauchte man,

Und was man weiss, kann man nicht brauchen.

Goethe, Faust I

One of the principal aims of this book is to describe some methods of constructing skew fields. The case most studied so far is that of skew fields finite-dimensional over their centres. But a finite-dimensional k-algebra, where k is a commutative field, is a field whenever it has no zero-divisors. On the one hand this enormously simplifies their study, while on the other hand it puts many constructions out of bounds (because they produce infinite-dimensional algebras). The study of fields that are not necessarily finite-dimensional over their centres is still in its early stages, and the methods needed here are not very closely related to those used on finite-dimensional algebras – the relation between these subjects is rather like the relation between finite and infinite groups.

There are some ways of obtaining a field directly, for example Schur's lemma tells us that the endomorphism ring of a simple module is a field, and the coordinatization theorem shows that when we coordinatize a Desarguesian plane, the coordinates lie in a field. But these methods are not very explicit, and we shall have no more to say about them. For us the usual way to construct a field is to take a suitable ring and embed it in a field.