##
**Inverse Galois theory.**
*(English)*
Zbl 0940.12001

Springer Monographs in Mathematics. Berlin: Springer. xv, 436 p. (1999).

Let \(K\) be a field, \(G\) a finite group. The inverse Galois theory asks whether there exists a (separable) polynomial \(f(X)\in K[X]\) with Galois group \(G\).

The first systematic approach for number fields goes back to Hilbert. During the last decades a lot of progress has been made. It is still unknown which finite groups occur as Galois groups over the field of rationals \(\mathbb{Q}\).

B. H. Matzat is one of the most active mathematicians in this area. In 1987 he presented his Springer lecture notes “Konstruktive Galoistheorie” (see the review in Zbl 0634.12011) which is mainly centered around the rigidity method. Together with his former student G. Malle he now publishes a new book on this subject. It covers earlier results, some with new proofs, as well as a large number of recent results in the inverse Galois theory.

During the last years the rigidity method has extensively been used to realize finite simple groups as Galois groups over \(\mathbb{Q}\). Several new versions of rigidity appeared in the literature.

The book is divided into five chapters: (I) The rigidity method. (II) Applications of rigidity. (III) Action of the braids. (IV) Embedding problems. (V) Rigid analytic methods.

In chapter I the authors explain the rigidity method for coverings of the projective line in characteristic zero. The basic rigidity theorem gives a criterion on a finite group \(G\) which guarantees that \(G\) is realizable as a Galois group over \(\mathbb{Q}_C(t)\), where \(\mathbb{Q}_C/\mathbb{Q}\) is a finite abelian extension. \(\mathbb{Q}_C\) is defined by certain character values of \(G\). In I.5 several applications of the rigidity method are given. It starts with abelian groups, the symmetric and the alternating group. Next we learn how to find rigid class vectors using character tables or suitable matrix representations. The consideration of the group of geometric automorphisms \(\operatorname{Aut} (\overline{\mathbb{Q}}(t)/ \overline{\mathbb{Q}})\) leads to a new twisted rigidity theorem, which enlarges the class of fields of definition of \(G\). Section I.8 gives some results on the realization of groups of automorphisms of \(G\) as a Galois group. Next the authors show how to compute generating polynomials for certain Galois groups by using the rigidity method. They illustrate this approach for the groups \({\mathfrak S}_n\), \({\mathfrak A}_n\), \(\operatorname{Aut} ({\mathfrak A}_6)\) and the Mathieu groups \({\mathfrak M}_{11}\), \({\mathfrak M}_{12}\).

The different kinds of rigidity theorems apply to a large class of finite groups. Specially the finite simple groups have been taken into account. The known verifications of the rigidity for the finite simple groups is presented in chapter II. The authors follow Belyi’s approach for the general linear groups. For the other classical groups they follow Walter’s approach, making use of an effective version of Belyi’s criterion due to Völklein.

The Deligne-Lusztig character theory for groups of Lie type is a main tool in the application of the rigidity method for exceptional groups of Lie type. The 26 sporadic simple groups except possible the Mathieu group \({\mathfrak M}_{23}\) are known to occur as Galois groups over \(\mathbb{Q}\). Several proofs of this result make use of computer calculations. The aim of II.9 is to avoid these calculations by using the informations of the atlas as far as possible. Chapter II closes with a summary of the known results on the realization of finite simple groups as Galois groups over \(\mathbb{Q}\) as well as over \(\mathbb{Q}^{ab}\).

The third chapter contains the development of the rigidity method for coverings of the projective space. By considering the action of the braids we get much weaker rigidity conditions as in the first chapter. Unfortunately, this only implies the regularity of the field of definition over \(\mathbb{Q}\). To apply Hilbert’s irreducibility theorem further arithmetic conditions like the existence of rational points has to be satisfied. The second part of chapter III contains the proof of Fried and Völklein that over functions fields \(k(t)\) with a PAC-field of constants \(k\) of characteristic zero every finite group occurs as a Galois group.

The first three chapters mainly center around the question of realizing finite simple groups as Galois groups. A nonsimple group \(\widetilde{G}\) contains a nontrivial normal subgroup \(H\) and a factor group \(G= \widetilde{G}/H\). How the theory of embedding problems asks under which conditions a realization of \(G\) as a Galois group over a given field \(K\) forces \(\widetilde{G}\) to occur as a Galois group over \(K\). Moreover we ask, whether a Galois extension \(N/K\) with Galois group \(G\) is contained in a Galois extension with Galois group \(\widetilde{G}\). After presenting the basic theory of embedding problems the authors consider embedding problems with abelian kernels and centerless embedding problems with GAR-kernels.

If the kernel \(H\) is abelian, the group extension \(\varphi: \widetilde{G}\to G\) defines a cohomology class \(h\) in \(H^2(G,H)\). It is well known that the image of \(h\) under the inflation map \(\varphi^*: H^2(G,H)\to H^2 (\Gamma_K,H)\) is the obstruction to the embedding problem under consideration. Here \(\varphi^*\) is defined by the Galois extension \(N/K\) with Galois group \(G\). \(\Gamma_K\) denotes the absolute Galois group of \(K\).

Section III.6 contains a cohomological approach to certain central embedding problems due to J.-P. Serre. Serre relates the obstruction to these embedding problems to invariants of the trace form of a given field extension. First, N. Vila used Serre’s trace form to realize the unique nontrivial double covers of the alternating group \({\mathfrak A}_n\) over \(\mathbb{Q}\) for a large class of natural number \(n\). J. F. Mestre generalized this approach to all \(n\) using Serre’s result. Later on, Völklein gave a new proof which avoids the use of trace form considerations. In section III.5 the authors show how the polynomial of Mestre and the result of Völklein can be used to prove that every central embedding problem of \({\mathfrak A}_n\) can be solved. Serre’s trace form considerations are illustrated with a result of J. Sonn on central group extensions of \({\mathfrak S}_n\) over \(\mathbb{Q}\).

III.8 and 9 investigate concordant embedding problems and the Hasse embedding obstruction. Here the authors start to touch the local-global theory of embedding problems. In 1937 Scholz and Reichardt proved the realizability of nilpotent groups as Galois groups over global fields. Section III.10 presents a new proof of this result due to Rzedowski-Calderon and Madan et al.

The first three chapters mainly consider the inverse Galois theory over \(\mathbb{Q}\), resp. \(\mathbb{Q}(t)\). Most of the results of the fourth chapter apply to a larger class of fields, for example to Hilbertian fields. The final chapter contains results on Galois groups over function fields of ultrametric fields of constants. Moreover, the authors study properties of the absolute Galois group of certain fields, for example \(\overline{\mathbb{F}}_p(t)\) and function fields of complete ultrametric fields.

In the appendix a large number of generating polynomials of Galois groups of small degree is given.

The first systematic approach for number fields goes back to Hilbert. During the last decades a lot of progress has been made. It is still unknown which finite groups occur as Galois groups over the field of rationals \(\mathbb{Q}\).

B. H. Matzat is one of the most active mathematicians in this area. In 1987 he presented his Springer lecture notes “Konstruktive Galoistheorie” (see the review in Zbl 0634.12011) which is mainly centered around the rigidity method. Together with his former student G. Malle he now publishes a new book on this subject. It covers earlier results, some with new proofs, as well as a large number of recent results in the inverse Galois theory.

During the last years the rigidity method has extensively been used to realize finite simple groups as Galois groups over \(\mathbb{Q}\). Several new versions of rigidity appeared in the literature.

The book is divided into five chapters: (I) The rigidity method. (II) Applications of rigidity. (III) Action of the braids. (IV) Embedding problems. (V) Rigid analytic methods.

In chapter I the authors explain the rigidity method for coverings of the projective line in characteristic zero. The basic rigidity theorem gives a criterion on a finite group \(G\) which guarantees that \(G\) is realizable as a Galois group over \(\mathbb{Q}_C(t)\), where \(\mathbb{Q}_C/\mathbb{Q}\) is a finite abelian extension. \(\mathbb{Q}_C\) is defined by certain character values of \(G\). In I.5 several applications of the rigidity method are given. It starts with abelian groups, the symmetric and the alternating group. Next we learn how to find rigid class vectors using character tables or suitable matrix representations. The consideration of the group of geometric automorphisms \(\operatorname{Aut} (\overline{\mathbb{Q}}(t)/ \overline{\mathbb{Q}})\) leads to a new twisted rigidity theorem, which enlarges the class of fields of definition of \(G\). Section I.8 gives some results on the realization of groups of automorphisms of \(G\) as a Galois group. Next the authors show how to compute generating polynomials for certain Galois groups by using the rigidity method. They illustrate this approach for the groups \({\mathfrak S}_n\), \({\mathfrak A}_n\), \(\operatorname{Aut} ({\mathfrak A}_6)\) and the Mathieu groups \({\mathfrak M}_{11}\), \({\mathfrak M}_{12}\).

The different kinds of rigidity theorems apply to a large class of finite groups. Specially the finite simple groups have been taken into account. The known verifications of the rigidity for the finite simple groups is presented in chapter II. The authors follow Belyi’s approach for the general linear groups. For the other classical groups they follow Walter’s approach, making use of an effective version of Belyi’s criterion due to Völklein.

The Deligne-Lusztig character theory for groups of Lie type is a main tool in the application of the rigidity method for exceptional groups of Lie type. The 26 sporadic simple groups except possible the Mathieu group \({\mathfrak M}_{23}\) are known to occur as Galois groups over \(\mathbb{Q}\). Several proofs of this result make use of computer calculations. The aim of II.9 is to avoid these calculations by using the informations of the atlas as far as possible. Chapter II closes with a summary of the known results on the realization of finite simple groups as Galois groups over \(\mathbb{Q}\) as well as over \(\mathbb{Q}^{ab}\).

The third chapter contains the development of the rigidity method for coverings of the projective space. By considering the action of the braids we get much weaker rigidity conditions as in the first chapter. Unfortunately, this only implies the regularity of the field of definition over \(\mathbb{Q}\). To apply Hilbert’s irreducibility theorem further arithmetic conditions like the existence of rational points has to be satisfied. The second part of chapter III contains the proof of Fried and Völklein that over functions fields \(k(t)\) with a PAC-field of constants \(k\) of characteristic zero every finite group occurs as a Galois group.

The first three chapters mainly center around the question of realizing finite simple groups as Galois groups. A nonsimple group \(\widetilde{G}\) contains a nontrivial normal subgroup \(H\) and a factor group \(G= \widetilde{G}/H\). How the theory of embedding problems asks under which conditions a realization of \(G\) as a Galois group over a given field \(K\) forces \(\widetilde{G}\) to occur as a Galois group over \(K\). Moreover we ask, whether a Galois extension \(N/K\) with Galois group \(G\) is contained in a Galois extension with Galois group \(\widetilde{G}\). After presenting the basic theory of embedding problems the authors consider embedding problems with abelian kernels and centerless embedding problems with GAR-kernels.

If the kernel \(H\) is abelian, the group extension \(\varphi: \widetilde{G}\to G\) defines a cohomology class \(h\) in \(H^2(G,H)\). It is well known that the image of \(h\) under the inflation map \(\varphi^*: H^2(G,H)\to H^2 (\Gamma_K,H)\) is the obstruction to the embedding problem under consideration. Here \(\varphi^*\) is defined by the Galois extension \(N/K\) with Galois group \(G\). \(\Gamma_K\) denotes the absolute Galois group of \(K\).

Section III.6 contains a cohomological approach to certain central embedding problems due to J.-P. Serre. Serre relates the obstruction to these embedding problems to invariants of the trace form of a given field extension. First, N. Vila used Serre’s trace form to realize the unique nontrivial double covers of the alternating group \({\mathfrak A}_n\) over \(\mathbb{Q}\) for a large class of natural number \(n\). J. F. Mestre generalized this approach to all \(n\) using Serre’s result. Later on, Völklein gave a new proof which avoids the use of trace form considerations. In section III.5 the authors show how the polynomial of Mestre and the result of Völklein can be used to prove that every central embedding problem of \({\mathfrak A}_n\) can be solved. Serre’s trace form considerations are illustrated with a result of J. Sonn on central group extensions of \({\mathfrak S}_n\) over \(\mathbb{Q}\).

III.8 and 9 investigate concordant embedding problems and the Hasse embedding obstruction. Here the authors start to touch the local-global theory of embedding problems. In 1937 Scholz and Reichardt proved the realizability of nilpotent groups as Galois groups over global fields. Section III.10 presents a new proof of this result due to Rzedowski-Calderon and Madan et al.

The first three chapters mainly consider the inverse Galois theory over \(\mathbb{Q}\), resp. \(\mathbb{Q}(t)\). Most of the results of the fourth chapter apply to a larger class of fields, for example to Hilbertian fields. The final chapter contains results on Galois groups over function fields of ultrametric fields of constants. Moreover, the authors study properties of the absolute Galois group of certain fields, for example \(\overline{\mathbb{F}}_p(t)\) and function fields of complete ultrametric fields.

In the appendix a large number of generating polynomials of Galois groups of small degree is given.

Reviewer: M.Epkenhans (Paderborn)

### MSC:

12F12 | Inverse Galois theory |

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

12F10 | Separable extensions, Galois theory |

12G05 | Galois cohomology |