Thinking Algebraically: An Introduction to Abstract Algebra

Length: 478 pages
Edition: 1
Language: English
Publisher: American Mathematical Society
Publication Date: 2021-04-08
ISBN-10: 1470460300
ISBN-13: 9781470460303
Sales Rank: #1374945 (See Top 100 Books)

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.\n\nThe 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.

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

Mathematics