Introduction

Summary

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.