# Algebra

## From the Viewpoint of Galois Theory

• Combines the standard material on algebraic field extensions with some related but advanced topics

• Each chapter starts with a motivating introduction and ends with a set of specially adapted exercises, some of them with solutions in the appendix

• Can be used as an accompanying textbook for a course, but also as a source for more specialized seminars

• The entire text is self-contained, up to a few rudimentary facts from Linear Algebra

Textbook

Part of the Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT)

1. Front Matter
Pages I-XV
2. Siegfried Bosch
Pages 9-22
3. Siegfried Bosch
Pages 23-81
4. Siegfried Bosch
Pages 83-132
5. Siegfried Bosch
Pages 133-229
6. Siegfried Bosch
Pages 231-253
7. Siegfried Bosch
Pages 255-282
8. Siegfried Bosch
Pages 283-321
9. Back Matter
Pages 323-367

### Introduction

The material presented here can be divided into two parts. The first, sometimes referred to as abstract algebra, is concerned with the general theory of algebraic objects such as groups, rings, and fields, hence, with topics that are also basic for a number of other domains in mathematics. The second centers around Galois theory and its applications. Historically, this theory originated from the problem of studying algebraic equations, a problem that, after various unsuccessful attempts to determine solution formulas in higher degrees, found its complete clarification through the brilliant ideas of E. Galois.

The study of algebraic equations has served as a motivating terrain for a large part of abstract algebra, and according to this, algebraic equations are visible as a guiding thread throughout the book. To underline this point, an introduction to the history of algebraic equations is included.

The entire book is self-contained, up to a few prerequisites from linear algebra. It covers most topics of current algebra courses and is enriched by several optional sections that complement the standard program or, in some cases, provide a first view on nearby areas that are more advanced. Every chapter begins with an introductory section on "Background and Overview," motivating the material that follows and discussing its highlights on an informal level. Furthermore, each section ends with a list of specially adapted exercises, some of them with solution proposals in the appendix.

The present English edition is a translation and critical revision of the eighth German edition of the Algebra book by the author. The book appeared for the first time in 1993 and, in later years, was complemented by adding a variety of related topics. At the same time it was modified and polished to keep its contents up to date.

### Keywords

algebraic equation field extension algebraic closure Euclid Euler Witt vectors irreducible polynomial splitting field alternating group profinite group

#### Authors and affiliations

1. 1.Mathematisches InstitutWestfälische Wilhelms-UniversitätMünsterGermany

Siegfried Bosch, Universität Münster, Germany.

### Bibliographic information

• Book Title Algebra
• Book Subtitle From the Viewpoint of Galois Theory
• Authors Siegfried Bosch
• Series Title Birkhäuser Advanced Texts Basler Lehrbücher
• Series Abbreviated Title Birkhäuser Advanced Texts(Birkhäuser)
• DOI https://doi.org/10.1007/978-3-319-95177-5
• Copyright Information Springer Nature Switzerland AG 2018
• Publisher Name Birkhäuser, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Hardcover ISBN 978-3-319-95176-8
• eBook ISBN 978-3-319-95177-5
• Series ISSN 1019-6242
• Series E-ISSN 2296-4894
• Edition Number 1
• Number of Pages XV, 352
• Number of Illustrations 13 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site