Since its inception at the beginning of the nineteenth century, the theory of field extensions has been a very active area of algebra. Its vitality stems not only from the interesting problems generated by the theory itself, but also from its connections with number theory and algebraic geometry. In writing this book, our principal objective has been to make the general theory of field extensions accessible to any reader with a modest background in groups, rings, and vector spaces.

The book is divided into four chapters. In order to give a precise idea of the background that the reader is expected to possess, we have preceded the text by a section on prerequisites. Except for the initial remarks, in which we indicate the restrictions that will be imposed on the rings considered throughout our presentation, the reader should not be concerned with the contents of this section until explicit reference is made to them. The first chapter is devoted to the general facts on fields and polynomials required in the study of field extensions. Although most of these facts can be found in one or another of the references given in the section on prerequisites, we have attempted to facilitate the reader's task by having them collected and stated in a manner suitably adapted to our purposes.

The theory of field extensions is presented in the subsequent three chapters, which deal, respectively, with algebraic extensions, Galois theory, and transcendental extensions.