Synopses & Reviews
Develops the mathematical background and recent results on the Inverse Galois Problem.
Review
"This book gives a comprehensible introduction to some aspects of Modern Galois theory....I highly recommend this book to all readers who would like to learn about this aspect of Galois theory, those who would like to give a course on inverse Galois theory and those who would like to see how different mathematical methods such as analysis, Riemann surface theory and group theory yield a nice algebraic result." Mathematical Reviews
Synopsis
This book is on the theory of symmetries in solutions of algebraic equations. It describes the Inverse Galois Problem, a classical unsolved problem posed by Hilbert at the beginning of the century, which brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. The author begins from the foundations and develops the necessary mathematical background to lead the reader to the research frontier. Graduate students and mathematicians from other areas will find this an excellent introduction to a fascinating field.
Table of Contents
Part 1. The Basic Rigidity Criteria: 1. Hilbert's irreducibility theorem; 2. Finite Galois extensions of C (x); 3. Descent of base field and the rigidity criterion; 4. Covering spaces and the fundamental group; 5. Riemann surfaces and their functional fields; 6. The analytic version of Riemann's existence theorem; Part II. Further Directions: 7. The descent from C to k; 8. Embedding problems: braiding action and weak rigidity; Moduli spaces for covers of the Riemann sphere; Patching over complete valued fields.