Groups as Galois Groups: An Introduction

by Helmut Volklein

Overview

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.

Product Details

ISBN-13:	9780521065030
Publisher:	Cambridge University Press
Publication date:	06/12/2008
Series:	Cambridge Studies in Advanced Mathematics , #53
Pages:	268
Product dimensions:	5.90(w) x 8.90(h) x 0.70(d)

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.

Editorial Reviews

"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

From the Publisher

For graduate students and mathematicians with other specialties, provides an understanding of recent results on the Inverse Galois Problem. Assuming only elementary algebra and complex analysis, develops the necessary space theory from topology, Riemann surface theory, and number theory. First establishes the basic rigidity criteria for realizing groups as Galois groups, then discusses more advanced topics such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, weak rigidity, and patching over complete valued fields. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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