Thinking Algebraically presents the insights of abstract algebra in a welcoming and accessible way. It succeeds in combining the advantages of rings-first and groups-first approaches while avoiding the disadvantages. After an historical overview, the first chapter studies familiar examples and elementary properties of groups and rings simultaneously to motivate the modern understanding of algebra. The text builds intuition for abstract algebra starting from high school algebra. In addition to the standard number systems, polynomials, vectors, and matrices, the first chapter introduces modular arithmetic and dihedral groups. The second chapter builds on these basic examples and properties, enabling students to learn structural ideas common to rings and groups: isomorphism, homomorphism, and direct product. The third chapter investigates introductory group theory. Later chapters delve more deeply into groups, rings, and fields, including Galois theory, and they also introduce other topics, such as lattices. The exposition is clear and conversational throughout.
The book has numerous exercises in each section as well as supplemental exercises and projects for each chapter. Many examples and well over 100 figures provide support for learning. Short biographies introduce the mathematicians who proved many of the results. The book presents a pathway to algebraic thinking in a semester- or year-long algebra course.
Author(s): Thomas Q. Sibley
Edition: 1
Publisher: American Mathematical Society
Year: 2021
Language: English
Commentary: Vector PDF
Pages: 478
City: Providence, RI
Tags: Mathematics; Abstract Algebra
Cover
Title page
Copyright
Contents
Preface
Topics
Features
Prologue
Exercises
Chapter 1. A Transitionto Abstract Algebra
1.1. An Historical View of Algebra
Exercises
1.2. Basic Algebraic Systems and Properties
Exercises
1.3. Functions, Symmetries,* and Modular Arithmetic
Exercises
Supplemental Exercises
Projects
Chapter 2. Relationshipsbetween Systems
2.1. Isomorphisms
Exercises
2.2. Elements and Subsets
Exercises
2.3. Direct Products
Exercises
2.4. Homomorphisms
Exercises
Supplemental Exercises
Projects
Chapter 3. Groups
3.1. Cyclic Groups
Exercises
3.2. Abelian Groups
Exercises
3.3. Cayley Digraphs
Exercises
3.4. Group Actions and Finite Symmetry Groups
Exercises
3.5. Permutation Groups, Part I
Exercises
3.6. Normal Subgroups and Factor Groups
Exercises
3.7. Permutation Groups, Part II
Exercises
Supplemental Exercises
Projects
Appendix: The Fundamental Theorem* of Finite Abelian Groups
Chapter 4. Rings, Integral Domains,and Fields
4.1. Rings and Integral Domains
Exercises
4.2. Ideals and Factor Rings
Exercises
4.3. Prime and Maximal Ideals
Exercises
4.4. Properties of Integral Domains
Exercises
4.5. Gröbner Bases in Algebraic Geometry
Exercises
4.6. Polynomial Dynamical Systems
Exercises
Supplemental Exercises
Projects
Chapter 5. Vector Spacesand Field Extensions
5.1. Vector Spaces
Exercises
5.2. Linear Codes and Cryptography
Exercises
5.3. Algebraic Extensions
Exercises
5.4. Geometric Constructions
Exercises
5.5. Splitting Fields
Exercises
5.6. Automorphisms of Fields
Exercises
5.7. Galois Theory* and the Insolvability of the Quintic
Exercises
Supplemental Exercises
Projects
Chapter 6. Topics in Group Theory
6.1. Finite Symmetry Groups
Exercises
6.2. Frieze, Wallpaper, and Crystal Patterns
Exercises
6.3. Matrix Groups
Exercises
6.4. Semidirect Products of Groups
Exercises
6.5. The Sylow Theorems
Exercises
Supplemental Exercises
Projects
Chapter 7. Topics in Algebra
7.1. Lattices and Partial Orders
Exercises
7.2. Boolean Algebras
Exercises
7.3. Semigroups
Exercises
7.4. Universal Algebra* and Preservation Theorems
Exercises
Supplemental Exercises
Projects
Epilogue
Selected Answers
Terms
Symbols
Names
Back Cover
