Generic polynomials. Constructive aspects of the inverse Galois problem. (English) Zbl 1042.12001

Jensen, Christian U.; Ledet, Arne; Yui, Noriko

Mathematical Sciences Research Institute Publications 45. Cambridge: Cambridge University Press (ISBN 0-521-81998-9/hbk). ix, 258 p. (2002).

Let \(G\) be a finite group and let \(K\) be a field. The inverse Galois theory asks whether there exists a Galois extension \(L/K\) of fields having Galois group \(G(L/K)\) isomorphic to \(G\). A constructive approach to this probem is to give an explicit polynomial over \(K\) whose Galois group is the prescribed group \(G\).
The main theme of the book under review is to find families of “generic” polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The book further introduces the notion of “generic dimension” to address the problem of the smallest number of parameters required by a generic polynomial.
The introduction contains some historical notes to inverse Galois theory and describes some techniques and strategies.
Chapter 2 deals with permutation groups of degree \(\leq 11\). It contains several well known criteria for the determination of the Galois group of an irreducible polynomial of degree \(\leq 11\). Some examples of generic polynomials are given.
Chapter 3 discusses Hilbertian fields and gives well known polynomials over \({\mathbb{Q}}(t)\) having Galois group the symmetric group resp. the alternating group.
In chapter 5 we find the definition and some basic results on generic \(G\)-extensions.
In the last chapters the authors study cyclic groups of odd order and several solvable groups such as \(p\)-groups and Frobenius groups.
Chapter 8 contains several results on the generic dimension.

Reviewer: Martin Epkenhans (Münster)

MSC:

12F12	Inverse Galois theory
12-02	Research exposition (monographs, survey articles) pertaining to field theory

Keywords:

Galois group; generic polynomial; inverse Galois theory