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A Classical Introduction to Galois Theory

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Explore the foundations and modern applications of Galois theory

Galois theory is widely regarded as one of the most elegant areas of mathematics. A Classical Introduction to Galois Theory develops the topic from a historical perspective, with an emphasis on the solvability of polynomials by radicals. The book provides a gradual transition from the computational methods typical of early literature on the subject to the more abstract approach that characterizes most contemporary expositions.

The author provides an easily-accessible presentation of fundamental notions such as roots of unity, minimal polynomials, primitive elements, radical extensions, fixed fields, groups of automorphisms, and solvable series. As a result, their role in modern treatments of Galois theory is clearly illuminated for readers. Classical theorems by Abel, Galois, Gauss, Kronecker, Lagrange, and Ruffini are presented, and the power of Galois theory as both a theoretical and computational tool is illustrated through:

  • A study of the solvability of polynomials of prime degree
  • Development of the theory of periods of roots of unity
  • Derivation of the classical formulas for solving general quadratic, cubic, and quartic polynomials by radicals

Throughout the book, key theorems are proved in two ways, once using a classical approach and then again utilizing modern methods. Numerous worked examples showcase the discussed techniques, and background material on groups and fields is provided, supplying readers with a self-contained discussion of the topic.

A Classical Introduction to Galois Theory is an excellent resource for courses on abstract algebra at the upper-undergraduate level. The book is also appealing to anyone interested in understanding the origins of Galois theory, why it was created, and how it has evolved into the discipline it is today.

Author(s): Stephen C. Newman
Edition: 1
Publisher: Wiley
Year: 2012

Language: English
Commentary: Bookmarks, cover, pagination
Pages: 297

Table of Contents

Cover

A Classical Introduction To Galois Theory

ISBN 9781118091395

Contents

Preface

Chapter 1 Classical Formulas

1.1 Quadratic Polynomials
1.2 Cubic Polynomials
1.3 Quartic Polynomials

Chapter 2 Polynomials And Field Theory

2.1 Divisibility
2.2 Algebraic Extensions
2.3 Degree Of Extensions
2.4 Derivatives
2.5 Primitive Element Theorem
2.6 Isomorphism Extension Theorem And Splitting Fields

Chapter 3 Fundamental Theorem On Symmetric Polynomials And Discriminants

3.1 Fundamental Theorem On Symmetric Polynomials
3.2 Fundamental Theorem On Symmetric Rational Functions
3.3 Some Identities Based On Elementary Symmetric Polynomials
3.4 Discriminants
3.5 Discriminants And Subfields Of The Real Numbers

Chapter 4 Irreducibility And Factorization

4.1 Irreducibility Over The Rational Numbers
4.2 Irreducibility And Splitting Fields
4.3 Factorization And Adjunction

Chapter 5 Roots Of Unity And Cyclotomic Polynomials

5.1 Roots Of Unity
5.2 Cyclotomic Polynomials

Chapter 6 Radical Extensions And Solvability By Radicals

6.1 Basic Results On Radical Extensions
6.2 Gauss's Theorem On Cyclotomic Polynomials
6.3 Abel's Theorem On Radical Extensions
6.4 Polynomials Of Prime Degree

Chapter 7 General Polynomials And The Beginnings Of Galois Theory

7.1 General Polynomials
7.2 The Beginnings Of Galois Theory

Chapter 8 Classical Galois Theory According To Galois

Chapter 9 Modern Galois Theory

9.1 Galois Theory And Finite Extensions
9.2 Galois Theory And Splitting Fields

Chapter 10 Cyclic Extensions And Cyclotomic Fields

10.1 Cyclic Extensions
10.2 Cyclotomic Fields

Chapter 11 Galois's Criterion For Solvability Of Polynomials By Radicals

Chapter 12 Polynomials Of Prime Degree

Chapter 13 Periods Of Roots Of Unity

Chapter 14 Denesting Radicals

Chapter 15 Classical Formulas Revisited

15.1 General Quadratic Polynomial
15.2 General Cubic Polynomial
15.3 General Quartic Polynomial

Appendix A Cosets And Group Actions

Appendix B Cyclic Groups

Appendix C Solvable Groups

Appendix D Permutation Groups

Appendix E Finite Fields And Number Theory

Appendix F Further Reading

References

Index