Galois Theory for Beginners: A Historical Perspective
About this Title
Jörg Bewersdorff. Translated by David Kramer
Publication: The Student Mathematical Library
Publication Year 2006: Volume 35
ISBNs: 978-0-8218-3817-4 (print); 978-1-4704-2146-5 (online)
MathSciNet review: MR2251389
MSC: Primary 12-01; Secondary 00-01, 01A05, 12F10
Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book, Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of particular investigations.
Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed. Results in Galois theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular $n$-gons are also presented.
This book is suitable for undergraduates and beginning graduate students.
Undergraduates and graduate students interested in Galois Theory.
Table of Contents
- Chapter 1. Cubic equations
- Chapter 2. Casus irreducibilis: The birth of the complex numbers
- Chapter 3. Biquadratic equations
- Chapter 4. Equations of degree $n$ and their properties
- Chapter 5. The search for additional solution formulas
- Chapter 6. Equations that can be reduced in degree
- Chapter 7. The construction of regular polygons
- Chapter 8. The solution of equations of the fifth degree
- Chapter 9. The Galois group of an equation
- Chapter 10. Algebraic structures and Galois theory