Abstract Algebra
Structures and Applications
Textbooks in Mathematics Series

A Discovery-Based Approach to Learning about Algebraic Structures

Abstract Algebra: Structures and Applications helps students understand the abstraction of modern algebra. It emphasizes the more general concept of an algebraic structure while simultaneously covering applications. The text can be used in a variety of courses, from a one-semester introductory course to a full two-semester sequence.

The book presents the core topics of structures in a consistent order:

• Definition of structure
• Motivation
• Examples
• General properties
• Important objects
• Description
• Subobjects
• Morphisms
• Subclasses
• Quotient objects
• Action structures
• Applications

The text uses the general concept of an algebraic structure as a unifying principle and introduces other algebraic structures besides the three standard ones (groups, rings, and fields). Examples, exercises, investigative projects, and entire sections illustrate how abstract algebra is applied to areas of science and other branches of mathematics.

 "Lovett (Wheaton College) takes readers through the variegated landscape of algebra, from elementary modular arithmetic through groups, semigroups, and monoids, past rings and fields and group actions, beyond modules and algebras, to Galois theory, multivariable polynomial rings, and Gröbner bases."

Choice Reviewed: Recommended

SET THEORY
Sets and Functions
The Cartesian Product; Operations; Relations
Equivalence Relations
Partial Orders

NUMBER THEORY
Basic Properties of Integers
Modular Arithmetic
Mathematical Induction

GROUPS
Symmetries of the Regular n-gon
Introduction to Groups
Properties of Group Elements
Symmetric Groups
Subgroups
Lattice of Subgroups
Group Homomorphisms
Group Presentations
Groups in Geometry
Diffie-Hellman Public Key
Semigroups and Monoids

QUOTIENT GROUPS
Cosets and Lagrange’s Theorem
Conjugacy and Normal Subgroups
Quotient Groups
Isomorphism Theorems
Fundamental Theorem of Finitely Generated Abelian Groups

RINGS
Introduction to Rings
Rings Generated by Elements
Matrix Rings
Ring Homomorphisms
Ideals
Quotient Rings
Maximal and Prime Ideals

DIVISIBILITY IN COMMUTATIVE RINGS
Divisibility in Commutative Rings
Rings of Fractions
Euclidean Domains
Unique Factorization Domains
Factorization of Polynomials
RSA Cryptography
Algebraic Integers

FIELD EXTENSIONS
Introduction to Field Extensions
Algebraic Extensions
Solving Cubic and Quartic Equations
Constructible Numbers
Cyclotomic Extensions
Splitting Fields and Algebraic Closures
Finite Fields

GROUP ACTIONS
Introduction to Group Actions
Orbits and Stabilizers
Transitive Group Actions
Groups Acting on Themselves
Sylow’s Theorem
A Brief Introduction to Representations of Groups

CLASSIFICATION OF GROUPS
Composition Series and Solvable Groups
Finite Simple Groups
Semidirect Product. Classification Theorems
Nilpotent Groups

MODULES AND ALGEBRAS
Boolean Algebras
Vector Spaces
Introduction to Modules
Homomorphisms and Quotient Modules
Free Modules and Module Decomposition
Finitely Generated Modules over PIDs, I
Finitely Generated Modules over PIDs, II
Applications to Linear Transformations
Jordan Canonical Form
Applications of the Jordan Canonical Form
A Brief Introduction to Path Algebras

GALOIS THEORY
Automorphisms of Field Extensions
Fundamental Theorem of Galois Theory
First Applications of Galois Theory
Galois Groups of Cyclotomic Extensions
Symmetries among Roots; The Discriminant
Computing Galois Groups of Polynomials
Fields of Finite Characteristic
Solvability by Radicals

MULTIVARIABLE POLYNOMIAL RINGS
Introduction to Noetherian Rings
Multivariable Polynomial Rings and Affine Space
The Nullstellensatz
Polynomial Division; Monomial Orders
Gröbner Bases
Buchberger’s Algorithm
Applications of Gröbner Bases
A Brief Introduction to Algebraic Geometry

CATEGORIES
Introduction toCategories
Functors

Projects appear at the end of each chapter.

Stephen Lovett is an associate professor of mathematics at Wheaton College. He is a member of the Mathematical Association of America, American Mathematical Society, and Association of Christians in the Mathematical Sciences. He earned a PhD from Northeastern University. His research interests include commutative algebra, algebraic geometry, differential geometry, cryptography, and discrete dynamical systems.