Inverse Galois Theory

Description

This second edition addresses the question of which finite groups occur as Galois groups over a given field. In particular, this includes the question of the structure and the representations of the absolute Galois group of K, as well as its finite epimorphic images, generally referred to as the inverse problem of Galois theory. In the past few years, important strides have been made in all of these areas. The aim of the book is to provide a systematic and extensive overview of these advances, with special emphasis on the rigidity method and its applications. Among others, the book presents the most successful known existence theorems and construction methods for Galois extensions and solutions of embedding problems, together with a collection of the current Galois realizations. There have been two major developments since the first edition of this book was released. The first is the algebraization of the Katz algorithm for (linearly) rigid generating systems of finite groups; the second is the emergence of a modular Galois theory. The latter has led to new construction methods for additive polynomials with given Galois group over fields of positive characteristic. Both methods have their origin in the Galois theory of differential and difference equations. Front Matter ....Pages i-xvii The Rigidity Method (Gunter Malle, B. Heinrich Matzat)....Pages 1-90 Applications of Rigidity (Gunter Malle, B. Heinrich Matzat)....Pages 91-175 Action of Braids (Gunter Malle, B. Heinrich Matzat)....Pages 177-284 Embedding Problems (Gunter Malle, B. Heinrich Matzat)....Pages 285-382 Additive Polynomials (Gunter Malle, B. Heinrich Matzat)....Pages 383-446 Rigid Analytic Methods (Gunter Malle, B. Heinrich Matzat)....Pages 447-489 Back Matter ....Pages 491-533