Groups as Galois Groups An Introduction
Groups as Galois Groups An Introduction by Helmut Volklein, published by Cambridge University Press on June 12, 2008, is a comprehensive exploration of the Inverse Galois Problem, a classical unsolved issue in mathematics. This edition spans 268 pages and is presented in English. The book integrates concepts from group theory, algebraic geometry, number theory, topology, and analysis, making it a valuable resource for those interested in these mathematical disciplines.
Readers will find that the book begins with elementary concepts, gradually leading to the fundamental rigidity criteria necessary for realizing groups as Galois groups. The latter sections delve into more complex topics, including braid group action and moduli spaces for covers of the Riemann sphere. The text is designed to be accessible to graduate students and mathematicians from various fields, particularly those focused on group theory, providing a solid introduction to this intriguing area of study.
Official synopsis Publisher
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.
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