# Abstract Algebra

## An Interactive Approach

#### By **William Paulsen**

#### Series Editor: **Denny Gulick**

CRC Press – 2009 – 560 pages

**Series:** Textbooks in Mathematics

CRC Press – 2009 – 560 pages

**Series:** Textbooks in Mathematics

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.

"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

**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.*

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

Name: Abstract Algebra: An Interactive Approach (Hardback) – CRC Press

Description: By William PaulsenSeries Editor: Denny Gulick. 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...

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