Galois Theory for Beginners

Front Cover
American Mathematical Soc., 2006 - Mathematics - 180 pages
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.
 

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Contents

Chapter 1 Cubic Equations
1
The Birth of the Complex Numbers
9
Chapter 3 Biquadratic Equations
23
Chapter 4 Equations of Degree n and Their Properties
27
Chapter 5 The Search for Additional Solution Formulas
37
Chapter 6 Equations That Can Be Reduced in Degree
55
Chapter 7 The Construction of Regular Polygons
63
Chapter 8 The Solution of Equations of the Fifth Degree
81
Chapter 9 The Galois Group of an Equation
93
Chapter 10 Algebraic Structures and Galois Theory
125
Epilogue
165
Index
177
Back Cover
181
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