Khovanskii, Askold.
Galois Theory, Coverings, and Riemann Surfaces [electronic resource] / by Askold Khovanskii. - VIII, 81 p. online resource.
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.
9783642388415
10.1007/978-3-642-38841-5 doi
Mathematics.
Algebra.
Geometry, algebraic.
Field theory (Physics).
Group theory.
Topology.
Mathematics.
Field Theory and Polynomials.
Group Theory and Generalizations.
Topology.
Algebra.
Algebraic Geometry.
QA161.A-161.Z QA161.P59
512.3
Galois Theory, Coverings, and Riemann Surfaces [electronic resource] / by Askold Khovanskii. - VIII, 81 p. online resource.
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.
9783642388415
10.1007/978-3-642-38841-5 doi
Mathematics.
Algebra.
Geometry, algebraic.
Field theory (Physics).
Group theory.
Topology.
Mathematics.
Field Theory and Polynomials.
Group Theory and Generalizations.
Topology.
Algebra.
Algebraic Geometry.
QA161.A-161.Z QA161.P59
512.3