Volklein H. Groups as Galois groups: An introduction

Cambridge University Press, 2008. — 267 p. — (Cambridge Studies in Advanced Mathematics 53). — ISBN: 0521065038.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.The Basic Rigidity Criteria
Finite Galois Extensions of C(x)
Descent of Base Field and the Rigidity Criterion
Covering Spaces and the Fundamental Group
Riemann Surfaces and Their Function Fields
The Analytic Version of Riemann's Existence Theorem
Further Directions
The Descent from C to k
Embedding Problems
Braiding Action and Weak Rigidity
Moduli Spaces for Covers of the Riemann Sphere
Patching over Complete Valued Fields