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.

Review of the hardback:‘I highly recommend this book to all readers who like to learn this aspect of Galois theory, those who like to give a course on Galois theory and those who like to see how different mathematical methods as analysis, Riemann surface theory and group theory yield a nice algebraic result.’

Translated from Martin Epkenhans Source: Zentralblatt für Mathematiche

Review of the hardback:‘… a very helpful introduction into an active research area, recommended for graduate students and anyone interested in recent progress in the inverse Galois problem.’

B. H. Matzat Source: Bulletin of London Mathmatical Society