Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms (Springer Monographs in Mathematics)

This book provides a detailed and largely self-contained description of various classical and new results on solvability and unsolvability of equations in explicit form. In particular, it offers a complete exposition of the relatively new area of topological Galois theory, initiated by the author. Applications of Galois theory to solvability of algebraic equations by radicals, basics of Picard–Vessiot theory, and Liouville's results on the class of functions representable by quadratures are also discussed. A unique feature of this book is that recent results are presented in the same elementary manner as classical Galois theory, which will make the book useful and interesting to readers with varied backgrounds in mathematics, from undergraduate students to researchers. In this English-language edition, extra material has been added (Appendices A–D), the last two of which were written jointly with Yura Burda. Front Matter....Pages i-xviii Construction of Liouvillian Classes of Functions and Liouville’s Theory....Pages 1-45 Solvability of Algebraic Equations by Radicals and Galois Theory....Pages 47-83 Solvability and Picard–Vessiot Theory....Pages 85-106 Coverings and Galois Theory....Pages 107-141 One-Dimensional Topological Galois Theory....Pages 143-171 Solvability of Fuchsian Equations....Pages 173-193 Multidimensional Topological Galois Theory....Pages 195-238 Back Matter....Pages 239-307 Askold Khovanskii. Based On Russian Edition Entitled Topologicheskaya Teoriya Galua, Razreshimost I Nerazreshimost Uravneii V Konechnom Vide, Published By Mccme, Moscow, Russia, 2008--title Page Verso. Includes Bibliographical References And Index.