##
**Polynomials with special regard to reducibility.**
*(English)*
Zbl 0956.12001

Encyclopedia of Mathematics and Its Applications. 77. Cambridge: Cambridge University Press. ix, 558 p. (2000).

This interesting and original monograph is devoted to questions of reducibility of polynomials defined over arbitrary fields. It contains much material that is unavailable elsewhere. As the author indicates in the introduction, of the 82 theorems proved in the text, 37 are original with the author and 23 have not appeared previously. The guiding principle is that, as far as possible, the results included are those valid over general fields. Successive chapters specialize the assumptions on the field in a natural way to obtain more precise results. The book contains a wealth of material that will reward careful study. At the end of each section there are interesting and informative historical and bibliographical notes that guide the reader to the available literature.

Chapter 1 has a complete account of Ritt’s theorems on decomposability of polynomials with their applications to primitivity of field extensions and Galois theory. The relationship between decomposability of \(f(x)\) and irreducibility of the two variable polynomial \((f(x)-f(y))/(x-y)\) is also found in this chapter, as is an extension to the reducibility of polynomials of the form \(\phi(F_1(x_1),\dots,F_n(x_n))\).

Chapter 2 treats the reducibility of lacunary polynomials over an arbitrary field. It begins with the most lacunary of polynomials, \(x^n - a\), giving a complete account of Kneser’s generalization of a classical theorem of Capelli on the reducibility of \(x^n - a\) over a field \(\mathbf k\) and a generalization due to the author concerned with the degree of multiple extensions by pure radicals. This is then applied to the question of the factorization of polynomials of the form \(F(x_1^{\nu_1},\dots,x_s^{\nu_s})\). Subsequent sections deal with the reducibility of trinomials and quadrinomials in many variables.

Chapter 3 treats polynomials over an algebraically closed field. A large portion of this chapter is devoted to the a treatment of the Mahler measure of polynomials in many variables over \(\mathbb C\). This is certainly the most complete treatment of Mahler’s measure in print and contains much original material.

Chapter 4 considers the factorization of many variable polynomials over a finitely generated field, again treating the factorization of \(F(x_1^{\nu_1},\dots,x_s^{\nu_s})\) using the tools developed in the previous chapters. Also included in this chapter is a proof of Hilbert’s irreducibility theorem over all infinite finitely generated fields. This theorem concerns the question of specialization of parameters in irreducible polynomials in many variables.

Chapter 5 is devoted mainly to the following question. Suppose that \(F \in \mathbb C[{\mathbf x,\mathbf t}]\), that \(\mathbf K\) is a number field and that \(F({\mathbf x},{\mathbf t}^*)\) has a zero in \({\mathbf K}^s\) for a sufficiently large set of \({\mathbf t}^* \in \mathbb Z^r\), then does it follow that \(F\) regarded as a polynomial in \(\mathbf x\) has a zero in \({\mathbf K(t)}^s\)?

Chapter 6 deals with polynomials over a Kroneckerian field, i.e. a totally real field or a totally complex quadratic extension of a totally real field (so that complex conjugation commutes with Galois conjugation). A long first section deals with the Mahler measure of non-self-inversive polynomials. Subsequent sections contain bounds on the number of non-self-inversive and the number of self-inversive factors of lacunary polynomials in many variables.

The book concludes with a number of appendices giving a useful summary of background material. There is also a long appendix by U. Zannier giving a proof of a conjecture from the text. There is a very thorough 15 page bibliography as well as a short but useful index of terminology in addition to the 4 page section on notation that begins the text.

Chapter 1 has a complete account of Ritt’s theorems on decomposability of polynomials with their applications to primitivity of field extensions and Galois theory. The relationship between decomposability of \(f(x)\) and irreducibility of the two variable polynomial \((f(x)-f(y))/(x-y)\) is also found in this chapter, as is an extension to the reducibility of polynomials of the form \(\phi(F_1(x_1),\dots,F_n(x_n))\).

Chapter 2 treats the reducibility of lacunary polynomials over an arbitrary field. It begins with the most lacunary of polynomials, \(x^n - a\), giving a complete account of Kneser’s generalization of a classical theorem of Capelli on the reducibility of \(x^n - a\) over a field \(\mathbf k\) and a generalization due to the author concerned with the degree of multiple extensions by pure radicals. This is then applied to the question of the factorization of polynomials of the form \(F(x_1^{\nu_1},\dots,x_s^{\nu_s})\). Subsequent sections deal with the reducibility of trinomials and quadrinomials in many variables.

Chapter 3 treats polynomials over an algebraically closed field. A large portion of this chapter is devoted to the a treatment of the Mahler measure of polynomials in many variables over \(\mathbb C\). This is certainly the most complete treatment of Mahler’s measure in print and contains much original material.

Chapter 4 considers the factorization of many variable polynomials over a finitely generated field, again treating the factorization of \(F(x_1^{\nu_1},\dots,x_s^{\nu_s})\) using the tools developed in the previous chapters. Also included in this chapter is a proof of Hilbert’s irreducibility theorem over all infinite finitely generated fields. This theorem concerns the question of specialization of parameters in irreducible polynomials in many variables.

Chapter 5 is devoted mainly to the following question. Suppose that \(F \in \mathbb C[{\mathbf x,\mathbf t}]\), that \(\mathbf K\) is a number field and that \(F({\mathbf x},{\mathbf t}^*)\) has a zero in \({\mathbf K}^s\) for a sufficiently large set of \({\mathbf t}^* \in \mathbb Z^r\), then does it follow that \(F\) regarded as a polynomial in \(\mathbf x\) has a zero in \({\mathbf K(t)}^s\)?

Chapter 6 deals with polynomials over a Kroneckerian field, i.e. a totally real field or a totally complex quadratic extension of a totally real field (so that complex conjugation commutes with Galois conjugation). A long first section deals with the Mahler measure of non-self-inversive polynomials. Subsequent sections contain bounds on the number of non-self-inversive and the number of self-inversive factors of lacunary polynomials in many variables.

The book concludes with a number of appendices giving a useful summary of background material. There is also a long appendix by U. Zannier giving a proof of a conjecture from the text. There is a very thorough 15 page bibliography as well as a short but useful index of terminology in addition to the 4 page section on notation that begins the text.

Reviewer: David W.Boyd (Vancouver)

### MSC:

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

12E05 | Polynomials in general fields (irreducibility, etc.) |

11R09 | Polynomials (irreducibility, etc.) |