Galois Theory for Beginners: A Historical Perspective, Second Edition (Student Mathematical Library, 95)

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 the angle, and the construction of regular $n$-gons are also presented. This new edition contains an additional chapter as well as twenty facsimiles of milestones of classical algebra. It is suitable for undergraduates and graduate students, as well as teachers and mathematicians seeking a historical and stimulating perspective on the field. Cover Title page Copyright Contents Preface to the English Edition Translator’s Note Prefaces to the German Editions Preface to the First German Edition Acknowledgments Preface to the Second German Edition Preface to the Third German Edition Preface to the Fourth German Edition Preface to the Fifth German Edition Preface to the Sixth German Edition Chapter 1. Cubic Equations Literature on Quadratic and Cubic Equations Exercises Chapter 2. Casus Irreducibilis: The Birth of the Complex Numbers Literature on the History of Complex Numbers Exercises Chapter 3. Biquadratic Equations Literature on Biquadratic Equations Exercises Chapter 4. Equations of Degree n and Their Properties The Fundamental Theorem of Algebra: Plausibility and Proof Literature on Equations of Degree n Exercises Chapter 5. The Search for Additional Solution Formulas Permutations The Fundamental Theorem on Symmetric Polynomials Ruffini and the General Equation of Fifth Degree Exercises Chapter 6. Equations That Can Be Reduced in Degree The Decomposition of Integer Polynomials Eisenstein’s Irreducibility Criterion Exercises Chapter 7. The Construction of Regular Polygons Constructions with Straightedge and Compass The Classical Construction Problems Exercises Chapter 8. The Solution of Equations of the Fifth Degree The Transformations of Tschirnhaus and of Bring and Jerrard Literature on Equations of the Fifth Degree Exercises Chapter 9. The Galois Group of an Equation Computing the Galois Group A Quick Course in Calculating with Polynomials Literature on Galois Groups and Their History Exercises Chapter 10. Algebraic Structures and Galois Theory Groups and Fields The Fundamental Theorem of Galois Theory: An Example The Unsolvability of the Classical Construction Problems Literature on Galois Theory and Algebraic Structures Exercises Chapter 11. Galois Theory According to Artin Literature on Galois Theory from the Viewpoint of Linear Algebra Exercises Epilogue Exercises Index Back Cover