Topological Galois Theory : Solvability and Unsolvability of Equations in Finite Terms 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 Liouvilles 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 Preface 1 Construction of Liouvillian Classes of Functions and Liouville’s Theory 2 Solvability of Algebraic Equations by Radicals and Galois Theory 3 Solvability and Picard Vessiot Theory 4 Coverings and Galois Theory 5 One Dimensional Topological Galois Theory 6 Solvability of Fuchsian Equations 7 Multidimensional Topological Galois Theory Appendix A: Straightedge and Compass Constructions Appendix B: Chebyshev Polynomials and Their Inverses Appendix C: Signatures of Branched Coverings and Solvability in Quadratures Appendix D: On an Algebraic Version of Hilbert’s 13th Problem References