Galois Theory for Beginners: A Historical Perspective

by Jörg Bewersdorff, David Kramer (Translator)

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.

Genres: Mathematics, Reference

180 pages, Paperback

First published April 1, 2002

Reviews

[Samuel · Rating 4 out of 5]

The usual presentation of Galois Theory misses some of the original insights and motivations from the problem of solubility by radicals. In this short book, we have the opportunity to appreciate step by step how these ideas have evolved into an abstract rigorous package of theorems.

[Bruno Oliveira]

O título for beginners as vezes sugere um nível de facilidade que às vezes não corresponde ao grau de dificuldade da obra. No capítulo sobre a solução das equações cúbicas a controvérsia Tartaglia-Cardano-Fiori-Del Ferro é tratada de uma perspectiva matemática um pouco diferente da apresentação tradicional. O capítulo 2 apresenta novas e interessantes perspectivas sobre os números complexos, valendo destacar ainda no capítulo dez a existência de interessantes observações sobre os problemas clássicos irresolúveis de construtibilidade (duplicação do cubo, quadratura do círculo e trissecção do ângulo com régua e compasso). O grande objetivo da obra, no entanto, que é apresentar uma prova simples da teoria de Galois e da sua consequencia na resolução das equações de quinto grau parecem deixar um pouco a desejar, o que talvez não seja culpa do autor mas talvez da dificuldade do tema. De especial interesse o capítulo sobre a construção de polígonos regulares (cyclotomic equations).

[Liquidlasagna]

Amazon review

Too terse for my needs

When I see "for Beginners" in the title of a math book, my expectation is that the proofs of theorems will contain many simple steps and build on well-formulated definitions. This book did not meet those expectations. The proofs were terse and hard to follow, and the nomenclature was stilted (probably a result from translating the original German prose). The first few chapters were fine, and at times even interesting, but from chapter 4 onward, the text became unreadable for a true beginner. Very disappointing, especially considering the book's steep price.

4/10

Michael J. Skarpelos