Topics in Galois theory, Second Edition

✍ Scribed by Jean-Pierre Serre

• Publisher: AK Peters
• Year: 2008
• Tongue: English
• Leaves: 134
• Edition: 2
• Category: Library

✦ Synopsis

This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Scholz and Reichardt construction for p-groups, p not equal 2, as well as Hilbert s irreducibility theorem and the large sieve inequality, are presented. The second half is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T). While proofs are not carried out in full detail, the book contains a number of examples, exercises, and open problems.

✦ Table of Contents

Contents......Page 6
Foreword......Page 10
Notation......Page 12
Introduction......Page 14
1.1 The groups Z/2Z, Z/3Z, and S3......Page 18
1.2 The group C4......Page 19
1.3 Application of tori to abelian Galois groups of exponent 2, 3, 4, 6......Page 23
2.1 A theorem of Scholz-Reichardt......Page 26
2.2 The Frattini subgroup of a finite group......Page 33
3.1 The Hilbert property......Page 36
3.2 Properties of thin sets......Page 38
3.3 Irreducibility theorem and thin sets......Page 40
3.4 Hilbert's irreducibility theorem......Page 42
3.5 Hilbert property and weak approximation......Page 45
3.6 Proofs of prop. 3.5.1 and 3.5.2......Page 48
4.1 The property GalT......Page 52
4.2 Abelian groups......Page 53
4.3 Example: the quaternion group Q8......Page 55
4.4 Symmetric groups......Page 56
4.5 The alternating group An......Page 60
4.6 Finding good specializations of T......Page 61
5.1 Statement of Shih's theorem......Page 64
5.2 An auxiliary construction......Page 65
5.3 Proof of Shih's theorem......Page 66
5.4 A complement......Page 69
5.5 Further results on PSL2(Fq) and SL2(Fq) as Galois groups......Page 70
6.1 The GAGA principle......Page 72
6.3 From C to Q......Page 74
6.4 Appendix: universal ramified coverings of Riemann surfaces with signature......Page 77
7.1 Rationality......Page 82
7.2 Counting solutions of equations in finite groups......Page 84
7.3 Rigidity of a family of conjugacy classes......Page 87
7.4 Examples of rigidity......Page 89
8.1 The main theorem......Page 98
8.2 Two variants......Page 101
8.3 Examples......Page 102
8.4 Local properties......Page 106
9.1 Preliminaries......Page 112
9.2 The quadratic form Tr (x2)......Page 115
9.3 Application to extensions with Galois group Ãn......Page 117
10.1 Statement of the theorem......Page 120
10.3 The Davenport-Halberstam theorem......Page 122
10.4 Combining the information......Page 124
Bibliography......Page 126