It is my pleasant duty to thank here for all the help I received in the preparation of this book.

Colin Day, Director of the University of Michigan Press has permitted me to reuse material from my book Selected Topics on Polynomials published by the Press in question.

Professors Francesco Amoroso, David W. Boyd, Pierre Dèbes, Kálmán Győry, Gerhard Turnwald and Umberto Zannier have on my request read parts of the book, corrected mistakes and suggested many improvements. Chapter 1, Sections 1–3 of Chapter 3 and Section 9 of Chapter 5 have been read by U. Zannier. He has also written a very important appendix ‘Proof of Conjecture 1’. Chapter 2 has been read by G. Turnwald, who has also made most useful comments on Appendix A. Section 4 of Chapter 3 has been read by D.W. Boyd, Sections 1, 2, 3 of Chapter 4 by F. Amoroso, Section 4 of Chapter 5 and Sections 1–8 of Chapter 5 by P. Dèbes, finally Chapter 6 by K. Győry. In addition the whole book has been generously proofread by Jadwiga Lewkowicz and Andrzej Mąkowski, and the beginning of Chapter 1 by Andrzej Kondracki. I have also profited by advice from Dr. Michael Zieve concerning Section 5 of Chapter 4, from Professors Dieter Geyer, David Masser and Peter Roquette concerning Section 4 of Chapter 4 and from Professors Zbigniew Ciesielski, Piotr Mankiewicz, Aleksander Pełczyński and Dr. Marcin Kuczma concerning Appendix G.

This book is an attempt to cover most of the results on reducibility of polynomials over fairly large classes of fields; results valid only over finite fields, local fields or the rational field have not been included. On the other hand, included are many topics of interest to the author that are not directly related to reducibility, e.g. Ritt's theory of composition of polynomials.

Here is a brief summary of the six chapters.

Chapter 1 (Arbitrary polynomials over an arbitrary field) begins with Lüroth's theorem (Sections 1 and 2). This theorem is nowadays usually presented with a short non-constructive proof, due to Steinitz. We give a constructive proof and present the consequences Lüroth's theorem has for subfields of transcendence degree 1 of fields of rational functions in several variables. The much more difficult problem of the minimal number of generators for subfields of transcendence degree greater than 1 belongs properly to algebraic geometry and here only references are given.

The next topic to be considered (Sections 3 and 4) originated with Ritt. Ritt 1922 gave a complete analysis of the behaviour of polynomials in one variable over C under composition. He called a polynomial prime if it is not the composition of two polynomials of lower degree and proved the two main results:

1. In every representation of a polynomial as the composition of prime polynomials the number of factors is the same and their degrees coincide up to a permutation.