Topics in Galois theory. Notes written by Henri Darmon.

*(English)*Zbl 0746.12001
Research Notes in Mathematics. 1. Boston, MA etc.: Jones and Bartlett Publishers. xvi, 117 p. (1992).

A brief account of the content of this book is given in its foreword: “These notes are based on “Topics in Galois Theory”, a course given by J.-P. Serre at Harvard University in the Fall semester of 1988 and written down by H. Darmon. The course focused on the inverse problem of Galois theory: the construction of field extensions having a given finite group \(G\) as Galois group, typically over \(\mathbb{Q}\) but also over fields such as \(\mathbb{Q}(T)\).

Chapter 1 discusses examples for certain groups \(G\) of small order. The method of Scholz and Reichardt, which works over \(\mathbb{Q}\) when \(G\) is a \(p\)-group of odd order, is given in Chapter 2. Chapter 3 is devoted to the Hilbert irreducibility theorem and its connection with weak approximation and the large sieve inequality. Chapters 4 and 5 describe methods for showing that \(G\) is the Galois group of a regular extension of \(\mathbb{Q}(T)\) (one then says that \(G\) has property \(\mathrm{Gal}_T\)). Elementary constructions (e.g. when \(G\) is a symmetric or alternating group) are given in Chapter 4, while the method of Shih, which works for \(G=\mathrm{PSL}_2(p)\) in some cases, is outlined in Chapter 5. Chapter 6 describes the GAGA principle and the relation between the topological and algebraic fundamental groups of complex curves. Chapters 7 and 8 are devoted to the rationality and rigidity criteria and their application to proving the property \(\mathrm{Gal}_T\) for certain groups (notably, many of the sporadic simple groups, including the Fischer-Griess Monster). The relation between the Hasse-Witt invariant of the quadratic form \(\operatorname{Tr}(x^2)\) and certain embedding problems is the topic of Chapter 9, and an application to showing that \(\tilde A_ n\) has property \(\mathrm{Gal}_T\) is given. An appendix (Chapter 10) gives a proof of the large sieve inequality used in Chapter 3.

The reader should be warned that most proofs only give the main ideas; details have been left out. Moreover, a number of relevant topics have been omitted, for lack of time (and understanding), namely: a) The theory of generic extensions, ...... b) Shafarevich’s theorem on the existence of extensions of \(\mathbb{Q}\) with a given solvable Galois group, ...... c) The Hurwitz schemes which parametrize extensions with a given Galois group and a given ramification structure, ...... d) The computation of explicit equations for extensions with Galois group \(\mathrm{PSL}_2(F_7)\), \(\mathrm{SL}_2(F_ 8)\), \(M_{11},\ldots,\) ...... e) Mestre’s results ...... on extensions of \(\mathbb{Q}(T)\) with Galois group \(6\cdot A_6\), \(6\cdot A_7\), and \(\mathrm{SL}_2(F_7)\).”

The text contains many exercises and references to the literature. The author makes also many side remarks which relate the subject to other interesting parts of mathematics. According to the reviewer’s opinion one of the many highlights of these research notes is the relation between the inverse problem of Galois theory and weak approximation. In order to explain this connection let \(K\) be a number field and let \(\Sigma_K\) denote the set of all places of \(K\) (including the archimedean ones). For \(v\in\Sigma_K\) let \(K_v\) denote the completion of \(K\) at \(v\). If \(V\) is an absolutely irreducible integral variety over \(K\) the set of \(K_v\)-rational points \(V(K_v)\) of \(V\) is naturally endowed with a \(K_v\)-topology which gives it the structure of a \(K_v\)-analytic space. One says that \(V\) has the weak approximation property for a finite set of places \(S\subset\Sigma_K\) if \(V(K)\) is dense in \(\prod_{v\in S} V(K_ v)\). \(V\) is said to have property WA if it satisfies the weak approximation property with respect to \(S\) for all finite \(S\subset\Sigma_K\). It is said to have property WWA if there exists a finite set of places \(S_0\) of \(K\) such that \(V\) has the weak approximation property with respect to \(S\subset\Sigma_K\) for all \(S\) with \(S\cap S_0=\emptyset\).

The author mentions a conjecture of J.-L. Colliot-Thélène, namely: Every \(K\)-unirational smooth variety has the WWA property; and then the author proves that the statement of this conjecture implies that every finite group is isomorphic to the Galois group of some Galois extension of \(\mathbb{Q}\).

In conclusion, this is a very stimulating text which, according to the variety of methods and results, will attract mathematicians working in group theory, number theory, algebraic geometry and complex analysis.

Chapter 1 discusses examples for certain groups \(G\) of small order. The method of Scholz and Reichardt, which works over \(\mathbb{Q}\) when \(G\) is a \(p\)-group of odd order, is given in Chapter 2. Chapter 3 is devoted to the Hilbert irreducibility theorem and its connection with weak approximation and the large sieve inequality. Chapters 4 and 5 describe methods for showing that \(G\) is the Galois group of a regular extension of \(\mathbb{Q}(T)\) (one then says that \(G\) has property \(\mathrm{Gal}_T\)). Elementary constructions (e.g. when \(G\) is a symmetric or alternating group) are given in Chapter 4, while the method of Shih, which works for \(G=\mathrm{PSL}_2(p)\) in some cases, is outlined in Chapter 5. Chapter 6 describes the GAGA principle and the relation between the topological and algebraic fundamental groups of complex curves. Chapters 7 and 8 are devoted to the rationality and rigidity criteria and their application to proving the property \(\mathrm{Gal}_T\) for certain groups (notably, many of the sporadic simple groups, including the Fischer-Griess Monster). The relation between the Hasse-Witt invariant of the quadratic form \(\operatorname{Tr}(x^2)\) and certain embedding problems is the topic of Chapter 9, and an application to showing that \(\tilde A_ n\) has property \(\mathrm{Gal}_T\) is given. An appendix (Chapter 10) gives a proof of the large sieve inequality used in Chapter 3.

The reader should be warned that most proofs only give the main ideas; details have been left out. Moreover, a number of relevant topics have been omitted, for lack of time (and understanding), namely: a) The theory of generic extensions, ...... b) Shafarevich’s theorem on the existence of extensions of \(\mathbb{Q}\) with a given solvable Galois group, ...... c) The Hurwitz schemes which parametrize extensions with a given Galois group and a given ramification structure, ...... d) The computation of explicit equations for extensions with Galois group \(\mathrm{PSL}_2(F_7)\), \(\mathrm{SL}_2(F_ 8)\), \(M_{11},\ldots,\) ...... e) Mestre’s results ...... on extensions of \(\mathbb{Q}(T)\) with Galois group \(6\cdot A_6\), \(6\cdot A_7\), and \(\mathrm{SL}_2(F_7)\).”

The text contains many exercises and references to the literature. The author makes also many side remarks which relate the subject to other interesting parts of mathematics. According to the reviewer’s opinion one of the many highlights of these research notes is the relation between the inverse problem of Galois theory and weak approximation. In order to explain this connection let \(K\) be a number field and let \(\Sigma_K\) denote the set of all places of \(K\) (including the archimedean ones). For \(v\in\Sigma_K\) let \(K_v\) denote the completion of \(K\) at \(v\). If \(V\) is an absolutely irreducible integral variety over \(K\) the set of \(K_v\)-rational points \(V(K_v)\) of \(V\) is naturally endowed with a \(K_v\)-topology which gives it the structure of a \(K_v\)-analytic space. One says that \(V\) has the weak approximation property for a finite set of places \(S\subset\Sigma_K\) if \(V(K)\) is dense in \(\prod_{v\in S} V(K_ v)\). \(V\) is said to have property WA if it satisfies the weak approximation property with respect to \(S\) for all finite \(S\subset\Sigma_K\). It is said to have property WWA if there exists a finite set of places \(S_0\) of \(K\) such that \(V\) has the weak approximation property with respect to \(S\subset\Sigma_K\) for all \(S\) with \(S\cap S_0=\emptyset\).

The author mentions a conjecture of J.-L. Colliot-Thélène, namely: Every \(K\)-unirational smooth variety has the WWA property; and then the author proves that the statement of this conjecture implies that every finite group is isomorphic to the Galois group of some Galois extension of \(\mathbb{Q}\).

In conclusion, this is a very stimulating text which, according to the variety of methods and results, will attract mathematicians working in group theory, number theory, algebraic geometry and complex analysis.

Reviewer: Hans Opolka (Braunschweig)

##### MSC:

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

12F12 | Inverse Galois theory |

11R32 | Galois theory |

12E25 | Hilbertian fields; Hilbert’s irreducibility theorem |

11N35 | Sieves |

20D08 | Simple groups: sporadic groups |

20D06 | Simple groups: alternating groups and groups of Lie type |

20F29 | Representations of groups as automorphism groups of algebraic systems |