Abstract Algebra: Structures and Applications (Hardback) book cover

Abstract Algebra

Structures and Applications

By Stephen Lovett

© 2015 – Chapman and Hall/CRC

720 pages | 152 B/W Illus.

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pub: 2015-07-17
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Description

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

Reviews

"… lucid, detailed, and versatile main text comes with a wealth of illustrating examples and very instructive exercises in each single section of the book, and each chapter ends with a section containing project ideas (and hints) to challenge the student to write her or his own investigative or expository papers on related topics. … an excellent introduction to the principles of abstract algebra for upper undergraduate and graduate students, and a valuable source for instructors likewise. No doubt, this text is a highly welcome addition to the already existing plethora of primers on abstract algebra in the mathematical literature."

Zentralblatt MATH 1323

"This is a text for a serious upper-level undergraduate course in abstract algebra. It adopts a ‘groups first’ approach to the subject, and, although it starts from scratch, winds up covering more than enough material to fill out two semesters. The topic coverage is very extensive for an undergraduate text … The author does an excellent job of balancing theory with applications. … The inclusion of all the topics described above and the large number of exercises, examples and projects make for an undeniably interesting text … a well-written book with interesting features … "

MAA Reviews, November 2015

Table of Contents

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

APPENDICES

LIST OF NOTATIONS

BIBLIOGRAPHY

INDEX

Projects appear at the end of each chapter.

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General