Abstract Algebra

Abstract Algebra: An Interactive Approach

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Features

  • Offers the option of using technology in the classroom by incorporating GAP and Mathematica commands
  • Discusses topics not covered in similar texts, such as semi-direct products and skew fields
  • Includes many diagrams produced by Mathematica to help students visualize difficult concepts, such as homomorphisms and permutations
  • Contains numerous homework problems of both the interactive and standard types
  • Provides a CD-ROM with GAP packages and Mathematica notebooks

Solutions manual available for qualifying instructors

Summary

By integrating the use of GAP and Mathematica®, Abstract Algebra: An Interactive Approach presents a hands-on approach to learning about groups, rings, and fields. Each chapter includes both GAP and Mathematica commands, corresponding Mathematica notebooks, traditional exercises, and several interactive computer problems that utilize GAP and Mathematica to explore groups and rings.

Although the book gives the option to use technology in the classroom, it does not sacrifice mathematical rigor. It covers classical proofs, such as Abel’s theorem, as well as many graduate-level topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik’s Cube®-like puzzles, and Wedderburn’s theorem. He also incorporates problem sequences that allow students to delve into interesting topics in depth, including Fermat’s two square theorem.

This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area.

Table of Contents

Understanding the Group Concept

Introduction to Groups

Modular Arithmetic

Prime Factorizations

The Definition of a Group

The Structure within a Group

Generators of Groups

Defining Finite Groups in Mathematica and GAP

Subgroups

Patterns within the Cosets of Groups

Left and Right Cosets

How to Write a Secret Message

Normal Subgroups

Quotient Groups

Mappings between Groups

Isomorphisms

Homomorphisms

The Three Isomorphism Theorems

Permutation Groups

Symmetric Groups

Cycles

Cayley’s Theorem

Numbering the Permutations

Building Larger Groups from Smaller Groups

The Direct Product

The Fundamental Theorem of Finite Abelian Groups

Automorphisms

Semi-Direct Products

The Search for Normal Subgroups

The Center of a Group

The Normalizer and Normal Closure Subgroups

Conjugacy Classes and Simple Groups

The Class Equation and Sylow’s Theorems

Solvable and Insoluble Groups

Subnormal Series and the Jordan–Hölder Theorem

Derived Group Series

Polycyclic Groups

Solving the Pyraminx™

Introduction to Rings

Groups with an Additional Operation

The Definition of a Ring

Entering Finite Rings into GAP and Mathematica

Some Properties of Rings

The Structure within Rings

Subrings

Quotient Rings and Ideals

Ring Isomorphisms

Homomorphisms and Kernels

Integral Domains and Fields

Polynomial Rings

The Field of Quotients

Complex Numbers

Ordered Commutative Rings

Unique Factorization

Factorization of Polynomials

Unique Factorization Domains

Principal Ideal Domains

Euclidean Domains

Finite Division Rings

Entering Finite Fields in Mathematica or GAP

Properties of Finite Fields

Cyclotomic Polynomials

Finite Skew Fields

The Theory of Fields

Vector Spaces

Extension Fields

Splitting Fields

Galois Theory

The Galois Group of an Extension Field

The Galois Group of a Polynomial in Q

The Fundamental Theorem of Galois Theory

Solutions of Polynomial Equations Using Radicals

Bibliography

Answers to Odd Problems

Index

Problems appear at the end of each chapter.

Author Bio(s)

William Paulsen is a Professor of Mathematics at Arkansas State University.

Editorial Reviews

"The textbook gives an introduction to algebra. The course includes the explanation on how to use the computer algebra systems GAP and Mathematica …The book can be used for an undergraduate-level course (chapter 1-4 and 9-12) or a second semester graduate-level course."
—Gerhard Pfister, Zentralblatt MATH 1173

Downloads / Updates

Resource OS Platform Updated Description Instructions
C4521_Download_Files.zip Cross Platform March 17, 2014

 
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