Galois Theory, Coverings, and Riemann Surfaces

Häftad, Engelska, 2016

AvAskold Khovanskii

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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.

Produktinformation

Utgivningsdatum2016-08-23
Mått155 x 235 x undefined mm
FormatHäftad
SpråkEngelska
Antal sidor81
FörlagSpringer-Verlag Berlin and Heidelberg GmbH & Co. KG
ISBN9783662519561
OriginaltitelTeoriya Galua, Nakrytiya i Rimanovy Poverkhnosti
ÖversättareVladlen Timorin, Valentina Kiritchenko

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Grupper och gruppteori inom Naturvetenskap och teknik
Algebra inom Naturvetenskap och teknik
Matematik inom Naturvetenskap och teknik
Topologi inom Naturvetenskap och teknik

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

From the reviews: "This book features generalizations and variations beyond Abel's theorem per se. ... This book is for those who appreciate concision, and remarkably, the author develops these extended results in full detail--all in a work just a fraction of the length of standard Galois theory textbooks. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty." (D. V. Feldman, Choice, Vol. 51 (10), June, 2014)