Groups as Galois Groups
Groups as Galois Groups
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 realisation 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.
- Describes recent results on the Inverse Galois Problem
- Is elementary enough to be understood by second-year graduate students
Reviews & endorsements
"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
Product details
February 2011Adobe eBook Reader
9780511835629
0 pages
0kg
6 b/w illus.
This ISBN is for an eBook version which is distributed on our behalf by a third party.
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.
