# 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

Textbook

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

1. Front Matter
Pages i-xvii
2. Juliusz Brzeziński
Pages 1-8
3. Juliusz Brzeziński
Pages 9-11
4. Juliusz Brzeziński
Pages 13-17
5. Juliusz Brzeziński
Pages 19-25
6. Juliusz Brzeziński
Pages 27-33
7. Juliusz Brzeziński
Pages 35-41
8. Juliusz Brzeziński
Pages 43-45
9. Juliusz Brzeziński
Pages 47-50
10. Juliusz Brzeziński
Pages 51-58
11. Juliusz Brzeziński
Pages 59-63
12. Juliusz Brzeziński
Pages 65-71
13. Juliusz Brzeziński
Pages 73-75
14. Juliusz Brzeziński
Pages 77-80
15. Juliusz Brzeziński
Pages 81-83
16. Juliusz Brzeziński
Pages 85-91
17. Juliusz Brzeziński
Pages 93-107
18. Juliusz Brzeziński
Pages 109-150
19. Juliusz Brzeziński
Pages 151-175
20. Juliusz Brzeziński
Pages 177-234

### Introduction

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.

### Keywords

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

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

• Book Title Galois Theory Through Exercises
• Authors Juliusz Brzeziński
• Series Title Springer Undergraduate Mathematics Series
• Series Abbreviated Title SUMS
• DOI https://doi.org/10.1007/978-3-319-72326-6
• Copyright Information Springer International Publishing AG, part of Springer Nature 2018
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-72325-9
• eBook ISBN 978-3-319-72326-6
• Series ISSN 1615-2085
• Series E-ISSN 2197-4144
• Edition Number 1
• Number of Pages XVII, 293
• Number of Illustrations 12 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“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)