Abstract Algebra: An Interactive Approach, Second Edition, 2nd Edition (Hardback) book cover

Abstract Algebra

An Interactive Approach, Second Edition, 2nd Edition

By William Paulsen

Chapman and Hall/CRC

619 pages | 41 B/W Illus.

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pub: 2016-02-12
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Description

The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use to create opportunities for interactive learning and computer use.

This new edition offers a more traditional approach offering additional topics to the primary syllabus placed after primary topics are covered. This creates a more natural flow to the order of the subjects presented. This edition is transformed by historical notes and better explanations of why topics are covered.

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.

Each chapter includes, corresponding Sage notebooks, traditional exercises, and several interactive computer problems that utilize Sage and Mathematica® to explore groups, rings, fields and additional topics.

This text does not sacrifice mathematical rigor. It covers classical proofs, such as Abel’s theorem, as well as many 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. The author also incorporates problem sequences that allow students to delve into interesting topics, including Fermat’s two square theorem.

Reviews

Praise for previous editions:

"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

Table of Contents

Preliminaries

Integer Factorization

Functions

Modular Arithmetic

Rational and Real Numbers

Understanding the Group Concept

Introduction to Groups

Modular Congruence

The Definition of a Group

The Structure within a Group

Generators of Groups

Defining Finite Groups in Sage

Subgroups

Patterns within the Cosets of Groups

Left and Right Cosets

Writing Secret Messages

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 PyraminxTM

Introduction to Rings

The Definition of a Ring

Entering Finite Rings into Sage

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 Sage

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

Applications of Galois Theory

Appendix: Sage vs. Mathematica®

Answers to Odd-Numbered Problems

Bibliography

About the Author

William Paulsen, PhD,professor of mathematics, Arkansas State University, USA

About the Series

Textbooks in Mathematics

Learn more…

Subject Categories

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