This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realization of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.
Author(s): Helmut Volklein
Series: Cambridge Studies in Advanced Mathematics 53
Publisher: Cambridge University Press
Year: 1996
Language: English
Pages: 268
Contents......Page 8
Preface......Page 14
Notation......Page 18
Part One: The Basic Rigidity Criteria......Page 20
1.1 Hilbertian Fields......Page 22
1.2 The Rational Field Is Hilbertian......Page 32
1.3 Algebraic Extensions of Hilbertian Fields......Page 40
2 Finite Galois Extensions of C(x)......Page 45
2.1 Extensions of Laurent Series Fields......Page 46
2.2 Extensions of k(x)......Page 51
3.1 Descent......Page 59
3.2 The Rigidity Criteria......Page 67
3.3 Rigidity and the Simple Groups......Page 70
4.1 The General Theory......Page 80
4.2 Coverings of the Punctured Sphere......Page 88
5.1 Riemann Surfaces......Page 103
5.2 The Compact Riemann Surface Arising from a Covering......Page 106
5.3 Constructing Generators of G(Ljcex»......Page 111
5.4 Digression: The Equivalence between Coveringsand Field Extensions......Page 113
6.1 Abstract Hilbert Spaces......Page 115
6.2 The Hilbert Spaces L2(D)......Page 119
6.3 Cocycles and Coboundaries......Page 124
6.4 Cocycles on a Disc......Page 126
6.5 A Finiteness Theorem......Page 129
Part Two: Further Directions......Page 136
7.1 Extensions of C(x) Unramified Outside a Given Finite Set......Page 138
7.2 Specializing the Coefficients of an Absolutely IrreduciblePolynomial......Page 140
7.3 The Descent from C to k......Page 142
7.4 The Minimal Field of Definition......Page 145
7.5 Embedding Problems over k(x)......Page 147
8.1 Generalities......Page 149
8.2 Wreath Products and Split Abelian Embedding Problems......Page 153
8.3 GAR-Realizations and GAL-Realizations......Page 160
9 Braiding Action and Weak Rigidity......Page 174
9.1 Certain Galois Groups Associated with a Weakly RigidRamification Type......Page 175
9.2 Combinatorial Computation of .6. via Braid Group Actionand the Resulting Outer Rigidity Criterion......Page 180
9.3 Construction of Weakly Rigid Tuples......Page 184
9.4 An Application of the Outer Rigidity Criterion......Page 188
10 Moduli Spaces for Covers of theRiemann Sphere......Page 197
10.1 The Topological Construction of the Moduli Spaces......Page 198
10.2 The Algebraic Structure of the Moduli Spaces......Page 218
10.3 Digression: The Inverse Galois Problem andRational Points on Moduli Spaces......Page 227
11 Patching over Complete Valued Fields......Page 232
11.1 Power Series over Complete Rings......Page 233
11.2 Rings of Converging Power Series......Page 237
11.3 GAGA......Page 241
11.4 Galois Groups over k(x)......Page 250
References......Page 262
Index......Page 266
