Galois Theory, Coverings, and Riemann Surfaces

• Main Author: Khovanskii, Askold. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2013.
• Subjects: Mathematics.; Algebra.; Geometry, algebraic.; Field theory (Physics).; Group theory.; Topology.; Field Theory and Polynomials.; Group Theory and Generalizations.; Algebraic Geometry.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-3-642-38841-5

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 th...

Table of Contents:

• 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.