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An Invitation to Abstract Algebra

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Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics.

To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.

The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can―and cannot―represent the roots of polynomials.

The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory.

The presentation includes the following features:

  • Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters.

  • The text can be used for a one, two, or three-term course.

  • Each new topic is motivated with a question.

  • A collection of projects appears in Chapter 23.

Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks―period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

Author(s): Steven J. Rosenberg
Series: Textbooks in Mathematics
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2021

Language: English
Pages: 392
Tags: Abstract Algebra;

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
Symbols
1. Review of Sets, Functions, and Proofs
1.1. Sets
1.1.1. Some Special Sets of Numbers
1.1.2. Describing a Set
1.1.3. Operations on Sets
1.2. Functions
1.3. Proofs
1.3.1. Logic
1.3.2. Proof Conventions
1.4. How to Read This Book
2. Introduction: A Number Game
2.1. A Game with Integers
2.2. A Bigger Game
2.3. Concluding Remarks
2.4. Exercises
3. Groups
3.1. Introduction
3.2. Binary Operations
3.3. Groups: Definition and Some Examples
3.4. First Results about Groups
3.5. Exercises
4. Subgroups
4.1. Groups Inside Groups
4.2. The Subgroup Generated by a Set
4.3. Exercises
5. Symmetry
5.1. What is Symmetry?
5.2. Dihedral Groups
5.3. Exercises
6. Free Groups
6.1. The Free Group Generated by a Set
6.2. Exercises
7. Group Homomorphisms
7.1. Relationships between Groups
7.2. Kernels: How Much Did We Lose?
7.3. Cosets
7.4. Quotient Groups
7.5. Exercises
8. Lagrange’s Theorem
8.1. Cosets and Partitions
8.2. The Size of Cosets
8.3. Reaping the Consequences
8.4. Exercises
9. Special Types of Homomorphisms
9.1. Isomorphisms
9.2. Automorphisms
9.3. Embeddings
9.4. Exercises
10. Making Groups
10.1. Introduction
10.2. A Quotient Engine
10.3. Room for Everyone Inside
10.4. Exercises
11. Rings
11.1. A New Type of Structure
11.2. Ring Fundamentals
11.3. Ring Homomorphisms, Ideals, and Quotient Rings
11.4. Exercises
12. Results on Commutative Rings
12.1. Introduction
12.2. Primes and Domains
12.3. The Ideal Generated by a Set
12.4. Fields and Maximal Ideals
12.5. Exercises
13. Vector Spaces
13.1. Introduction
13.2. Abstract Vector Spaces
13.3. Bases: Generalized Coordinate Systems
13.4. Exercises
14. Polynomial Rings
14.1. Polynomials Over a Commutative Ring
14.2. Polynomials Over a Field
14.3. Exercises
15. Field Theory
15.1. Extension Fields
15.2. Splitting Fields
15.3. Exercises
16. Galois Theory
16.1. Field Embeddings
16.2. Separable Extensions
16.3. Normal Extensions
16.4. Galois Extensions
16.5. Exercises
17. Direct Sums and Direct Products
17.1. Introduction
17.2. Direct Products
17.3. Direct Sums
17.4. Exercises
18. The Structure of Finite Abelian Groups
18.1. Introduction
18.2. Preliminaries
18.3. Splitting into p-Subgroups
18.4. Structure of Abelian p-Groups
18.5. The Fundamental Theorem
18.6. Exercises
19. Group Actions
19.1. Groups Acting on Sets
19.2. Reaping the Consequences
19.3. Exercises
20. Learning from Z
20.1. Introduction
20.2. Fractions
20.3. Unique Factorization
20.4. Exercises
21. The Problems of the Ancients
21.1. Introduction
21.2. Constructible Numbers
21.3. Constructible Regular Polygons
21.4. Exercises
22. Solvability of Polynomial Equations by Radicals
22.1. Radicals
22.2. Solvable Polynomials
22.3. Solvable Groups
22.4. Galois Groups in the Generic Case
22.5. Which Groups Are Solvable?
22.6. The Grand Finale
22.7. Exercises
23. Projects
23.1. Gyrogroups
23.2. Kaleidoscopes
23.3. The Axiom of Choice
23.4. Some Category Theory
23.5. Linear Algebra: Change of Basis
23.6. Linear Algebra: Determinants
23.7. Linear Algebra: Eigenvalues
23.8. Linear Algebra: Rotations
23.9. Power Series
23.10. Quadratic Probing
23.11. Euclidean Domains
23.12. Resultants
23.13. Perfect Numbers and Lucas’s Test
23.14. Modules
Bibliography
Index