A Classical Introduction to Galois Theory

Description

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 degreeDevelopment of the theory of periods of roots of unityDerivation 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. Content: Chapter 1 Classical Formulas (pages 1–14): Chapter 2 Polynomials and Field Theory (pages 15–40): Chapter 3 Fundamental Theorem on Symmetric Polynomials and Discriminants (pages 41–64): Chapter 4 Irreducibility and Factorization (pages 65–79): Chapter 5 Roots of Unity and Cyclotomic Polynomials (pages 80–88): Chapter 6 Radical Extensions and Solvability by Radicals (pages 89–116): Chapter 7 General Polynomials and the Beginnings of Galois Theory (pages 117–134): Chapter 8 Classical Galois Theory According to Galois (pages 135–150): Chapter 9 Modern Galois Theory (pages 151–170): Chapter 10 Cyclic Extensions and Cyclotomic Fields (pages 171–184): Chapter 11 Galois's Criterion for Solvability of Polynomials by Radicals (pages 185–191): Chapter 12 Polynomials of Prime Degree (pages 192–199): Chapter 13 Periods of Roots of Unity (pages 200–224): Chapter 14 Denesting Radicals (pages 225–230): Chapter 15 Classical Formulas Revisited (pages 231–244):