© 2018

Galois Theory Through Exercises

  • Provides a hands-on approach to learning Galois theory, focusing on problem-solving exercises

  • Features almost 500 exercises with hints, answers or solutions

  • Includes Maple tutorials and exercises


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Back Matter
    Pages 235-293

About this book


This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises).

In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading.

A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.


Galois theory Galois theory exercises Galois theory computer-assisted examples cubic and quartic equations finite fields cyclotomic fields Galois resolvents lunes of Hippocrates inverse Galois problem solving algebraic equations of low degrees field extensions zeros of polynomials algebraic field extensions automorphism groups of fields Galois groups of finite field extensions Galois extensions Galois modules Solvability of equations

Authors and affiliations

  1. 1.Department of Mathematical SciencesUniversity of GothenburgSweden

About the authors

Juliusz Brzeziński is Professor Emeritus at the Department of Mathematical Sciences, which is a part of the University of Gothenburg and the Chalmers University of Technology, Sweden. His research concentrates on interactions between number theory, algebra and geometry of orders in algebras over global fields, in particular, in quaternion algebras. He is also interested in experimental number theory.

Bibliographic information


“This book contains a collection of exercises in Galois theory. … The book provides the readers with a solid exercise-based introduction to classical Galois theory; it will be useful for self-study or for supporting a lecture course.” (Franz Lemmermeyer, zbMATH 1396.12001, 2018)