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Classical Introduction to Galois Theory

ISBN-10: 1118091396

ISBN-13: 9781118091395

Edition: 2012

Authors: Stephen C. Newman

List price: $85.95
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Description:

This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals.  Both classical and modern approaches to the subject are described in turn in order to have the former (which is relatively concrete and computational) provide motivation for the latter (which can be quite abstract).  The theme of the book is historically the reason that Galois theory was created, and it continues to provide a platform for exploring both classical and modern concepts.  This book examines a number of problems arising in the area of classical mathematics, and a fundamental question to be considered is: For a given polynomial equation (over a given field), does a solution in terms of radicals exist?  That the need to investigate the very existence of a solution is perhaps surprising and invites an overview of the history of mathematics.  The classical material within the book includes theorems on polynomials, fields, and groups due to such luminaries as Gauss, Kronecker, Lagrange, Ruffini and, of course, Galois. These results figured prominently in earlier expositions of Galois theory, but seem to have gone out of fashion. This is unfortunate since, aside from being of intrinsic mathematical interest, such material provides powerful motivation for the more modern treatment of Galois theory presented later in the book.  Over the course of the book, three versions of the Impossibility Theorem are presented: the first relies entirely on polynomials and fields, the second incorporates a limited amount of group theory, and the third takes full advantage of modern Galois theory. This progression through methods that involve more and more group theory characterizes the first part of the book. The latter part of the book is devoted to topics that illustrate the power of Galois theory as a computational tool, but once again in the context of solvability of polynomial equations by radicals.
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Book details

List price: $85.95
Copyright year: 2012
Publisher: John Wiley & Sons Canada, Limited
Publication date: 7/17/2012
Binding: Hardcover
Pages: 296
Size: 6.50" wide x 9.50" long x 1.00" tall
Weight: 1.342
Language: English

Preface
Classical Formulas
Quadratic Polynomials
Cubic Polynomials
Quartic Polynomials
Polynomials and Field Theory
Divisibility
Algebraic Extensions
Degree of Extensions
Derivatives
Primitive Element Theorem
Isomorphism Extension Theorem and Splitting Fields
Fundamental Theorem on Symmetric Polynomials and Discriminants
Fundamental Theorem on Symmetric Polynomials
Fundamental Theorem on Symmetric Rational Functions
Some Identities Based on Elementary Symmetric Polynomials
Discriminants
Discriminants and Subfields of the Real Numbers
Irreducibility and Factorization
Irreducibility Over the Rational Numbers
Irreducibility and Splitting Fields
Factorization and Adjunction
Roots of Unity and Cyclotomic Polynomials
Roots of Unity
Cyclotomic Polynomials
Radical Extensions and Solvability by Radicals
Basic Results on Radical Extensions
Gauss's Theorem on Cyclotomic Polynomials
Abel's Theorem on Radical Extensions
Polynomials of Prime Degree
General Polynomials and the Beginnings of Galois Theory
General Polynomials
The Beginnings of Galois Theory
Classical Galois Theory According to Galois
Modern Galois Theory
Galois Theory and Finite Extensions
Galois Theory and Splitting Fields
Cyclic Extensions and Cyclotomic Fields
Cyclic Extensions
Cyclotomic Fields
Galois's Criterion for Solvability of Polynomials by Radicals
Polynomials of Prime Degree
Periods of Roots of Unity
Denesting Radicals
Classical Formulas Revisited
General Quadratic Polynomial
General Cubic Polynomial
General Quartic Polynomial
Cosets and Group Actions
Cyclic Groups
Solvable Groups
Permutation Groups
Finite Fields and Number Theory
Further Reading
References
Index