Galois theory, coverings, and Riemann surfaces / Askold Khovanskii.

Includes bibliographical references and index

Chapter 1 Galois Theory:
1.1 Action of a Solvable Group and Representability by Radicals.-
1.2 Fixed Points under an Action of a Finite Group and Its Subgroups.-
1.3 Field Automorphisms and Relations between Elements in a Field.-
1.4 Action of a k-Solvable Group and Representability by k-Radicals.-
1.5 Galois Equations.-
1.6 Automorphisms Connected with a Galois Equation.-
1.7 The Fundamental Theorem of Galois Theory.-
1.8 A Criterion for Solvability of Equations by Radicals.-
1.9 A Criterion for Solvability of Equations by k-Radicals.-
1.10 Unsolvability of Complicated Equations by Solving Simpler Equations.-
1.11 Finite Fields.-

Chapter 2 Coverings:
2.1 Coverings over Topological Spaces.-
2.2 Completion of Finite Coverings over Punctured Riemann Surfaces.-

Chapter 3 Ramified Coverings and Galois Theory:
3.1 Finite Ramified Coverings and Algebraic Extensions of Fields of Meromorphic Functions.-
3.2 Geometry of Galois Theory for Extensions of a Field of Meromorphic Functions.-

References.-
Index-

The first part of this book provides an elementary and self-contained exposition of classical Galois theory and its applications to questions of solvability of algebraic equations in explicit form. The second part describes a surprising analogy between the fundamental theorem of Galois theory and the classification of coverings over a topological space. The third part contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface and provides an introduction to the topological Galois theory developed by the author. All results are presented in the same elementary and self-contained manner as classical Galois theory, making this book both useful and interesting to readers with a variety of backgrounds in mathematics, from advanced undergraduate students to researchers.