Abstract Algebra: An Inquiry Based Approach, 1st Edition (Hardback) book cover

Abstract Algebra

An Inquiry Based Approach, 1st Edition

By Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom

Chapman and Hall/CRC

595 pages | 31 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9781466567061
pub: 2013-12-21
$125.00
x
eBook (VitalSource) : 9780429099991
pub: 2013-12-21
from $62.50


FREE Standard Shipping!

Description

To learn and understand mathematics, students must engage in the process of doing mathematics. Emphasizing active learning, Abstract Algebra: An Inquiry-Based Approach not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think.

The book can be used in both rings-first and groups-first abstract algebra courses. Numerous activities, examples, and exercises illustrate the definitions, theorems, and concepts. Through this engaging learning process, students discover new ideas and develop the necessary communication skills and rigor to understand and apply concepts from abstract algebra. In addition to the activities and exercises, each chapter includes a short discussion of the connections among topics in ring theory and group theory. These discussions help students see the relationships between the two main types of algebraic objects studied throughout the text.

Encouraging students to do mathematics and be more than passive learners, this text shows students that the way mathematics is developed is often different than how it is presented; that definitions, theorems, and proofs do not simply appear fully formed in the minds of mathematicians; that mathematical ideas are highly interconnected; and that even in a field like abstract algebra, there is a considerable amount of intuition to be found.

Reviews

"This book arose from the authors’ approach to teaching abstract algebra. They place an emphasis on active learning and on developing students’ intuition through their investigation of examples. … The text is organized in such a way that it is possible to begin with either rings or groups."

—Florentina Chirtes, Zentralblatt MATH 1295

Table of Contents

The Integers

The Integers: An Introduction

Introduction

Integer Arithmetic

Ordering Axioms

What’s Next

Concluding Activities

Exercises

Divisibility of Integers

Introduction

Quotients and Remainders

TheWell-Ordering Principle

Proving the Division Algorithm

Putting It All Together

Congruence

Concluding Activities

Exercises

Greatest Common Divisors

Introduction

Calculating Greatest Common Divisors

The Euclidean Algorithm

GCDs and Linear Combinations

Well-Ordering, GCDs, and Linear Combinations

Concluding Activities

Exercises

Prime Factorization

Introduction

Defining Prime

The Fundamental Theorem of Arithmetic

Proving Existence

Proving Uniqueness

Putting It All Together

Primes and Irreducibles in Other Number Systems

Concluding Activities

Exercises

Other Number Systems

Equivalence Relations and Zn

Congruence Classes

Equivalence Relations

Equivalence Classes

The Number System Zn

Binary Operations

Zero Divisors and Units in Zn

Concluding Activities

Exercises

Algebra

Introduction

Subsets of the Real Numbers

The Complex Numbers

Matrices

Collections of Sets

Putting It All Together

Concluding Activities

Exercises

Rings

An Introduction to Rings

Introduction

Basic Properties of Rings

Commutative Rings and Rings with Identity

Uniqueness of Identities and Inverses

Zero Divisors and Multiplicative Cancellation

Fields and Integral Domains

Concluding Activities

Exercises

Connections

Integer Multiples and Exponents

Introduction

Integer Multiplication and Exponentiation

Nonpositive Multiples and Exponents

Properties of Integer Multiplication and Exponentiation

The Characteristic of a Ring

Concluding Activities

Exercises

Connections

Subrings, Extensions, and Direct Sums

Introduction

The Subring Test

Subfields and Field Extensions

Direct Sums

Concluding Activities

Exercises

Connections

Isomorphism and Invariants

Introduction

Isomorphisms of Rings

Proving Isomorphism

Disproving Isomorphism

Invariants

Concluding Activities

Exercises

Connections

Polynomial Rings

Polynomial Rings

Polynomial Rings

Polynomials over an Integral Domain

Polynomial Functions

Concluding Activities

Exercises

Connections

Appendix – Proof that R[x] Is a Commutative Ring

Divisibility in Polynomial Rings

Introduction

The Division Algorithm in F[x]

Greatest Common Divisors of Polynomials

Relatively Prime Polynomials

The Euclidean Algorithm for Polynomials

Concluding Activities

Exercises

Connections

Roots, Factors, and Irreducible Polynomials

Polynomial Functions and Remainders

Roots of Polynomials and the Factor Theorem

Irreducible Polynomials

Unique Factorization in F[x]

Concluding Activities

Exercises

Connections

Irreducible Polynomials

Introduction

Factorization in C[x]

Factorization in R[x]

Factorization in Q[x]

Polynomials with No Linear Factors in Q[x]

Reducing Polynomials in Z[x] Modulo Primes

Eisenstein’s Criterion

Factorization in F[x] for Other Fields F

Summary

The Cubic Formula

Concluding Activities

Exercises

Appendix – Proof of the Fundamental Theorem of Algebra

Quotients of Polynomial Rings

Introduction

CongruenceModulo a Polynomial

Congruence Classes of Polynomials

The Set F[x]/hf(x)i

Special Quotients of Polynomial Rings

Algebraic Numbers

Concluding Activities

Exercises

Connections

More Ring Theory

Ideals and Homomorphisms

Introduction

Ideals

CongruenceModulo an Ideal

Maximal and Prime Ideals

Homomorphisms

The Kernel and Image of a Homomorphism

The First Isomorphism Theorem for Rings

Concluding Activities

Exercises

Connections

Divisibility and Factorization in Integral Domains

Introduction

Divisibility and Euclidean Domains

Primes and Irreducibles

Unique Factorization Domains

Proof 1: Generalizing Greatest Common Divisors

Proof 2: Principal Ideal Domains

Concluding Activities

Exercises

Connections

From Z to C

Introduction

FromW to Z

Ordered Rings

From Z to Q

Ordering on Q

From Q to R

From R to C

A Characterization of the Integers

Concluding Activities

Exercises

Connections

VI Groups 269

Symmetry

Introduction

Symmetries

Symmetries of Regular Polygons

Concluding Activities

Exercises

An Introduction to Groups

Groups

Examples of Groups

Basic Properties of Groups

Identities and Inverses in a Group

The Order of a Group

Groups of Units

Concluding Activities

Exercises

Connections

Integer Powers of Elements in a Group

Introduction

Powers of Elements in a Group

Concluding Activities

Exercises

Connections

Subgroups

Introduction

The Subgroup Test

The Center of a Group

The Subgroup Generated by an Element

Concluding Activities

Exercises

Connections

Subgroups of Cyclic Groups

Introduction

Subgroups of Cyclic Groups

Properties of the Order of an Element

Finite Cyclic Groups

Infinite Cyclic Groups

Concluding Activities

Exercises

The Dihedral Groups

Introduction

Relationships between Elements in Dn

Generators and Group Presentations

Concluding Activities

Exercises

Connections

The Symmetric Groups

Introduction

The Symmetric Group of a Set

Permutation Notation and Cycles

The Cycle Decomposition of a Permutation

Transpositions

Even and Odd Permutations and the Alternating Group

Concluding Activities

Exercises

Connections

Cosets and Lagrange’s Theorem

Introduction

A Relation in Groups

Cosets

Lagrange’s Theorem

Concluding Activities

Exercises

Connections

Normal Subgroups and Quotient Groups

Introduction

An Operation on Cosets

Normal Subgroups

Quotient Groups

Cauchy’s Theorem for Finite Abelian Groups

Simple Groups and the Simplicity of An

Concluding Activities

Exercises

Connections

Products of Groups

External Direct Products of Groups

Orders of Elements in Direct Products

Internal Direct Products in Groups

Concluding Activities

Exercises

Connections

Group Isomorphisms and Invariants

Introduction

Isomorphisms of Groups

Proving Isomorphism

Some Basic Properties of Isomorphisms

Well-Defined Functions

Disproving Isomorphism

Invariants

Isomorphism Classes

Isomorphisms and Cyclic Groups

Cayley’s Theorem

Concluding Activities

Exercises

Connections

Homomorphisms and Isomorphism Theorems

Homomorphisms

The Kernel of a Homomorphism

The Image of a Homomorphism

The Isomorphism Theorems for Groups

Concluding Activities

Exercises

Connections

The Fundamental Theorem of Finite Abelian Groups

Introduction

The Components: p-Groups

The Fundamental Theorem

Concluding Activities

Exercises

The First Sylow Theorem

Introduction

Conjugacy and the Class Equation

Cauchy’s Theorem

The First Sylow Theorem

The Second and Third Sylow Theorems

Concluding Activities

Exercises

Connections

The Second and Third Sylow Theorems

Introduction

Conjugate Subgroups and Normalizers

The Second Sylow Theorem

The Third Sylow Theorem

Concluding Activities

Exercises

Special Topics

RSA Encryption

Introduction

Congruence and Modular Arithmetic

The Basics of RSA Encryption

An Example

Why RSA Works

Concluding Thoughts and Notes

Exercises

Check Digits

Introduction

Check Digits

Credit Card Check Digits

ISBN Check Digits

Verhoeff’s Dihedral Group D5 Check

Concluding Activities

Exercises

Connections

Games: NIM and the 15 Puzzle

The Game of NIM

The 15 Puzzle

Concluding Activities

Exercises

Connections

Finite Fields, the Group of Units in Zn, and Splitting Fields

Introduction

Finite Fields

The Group of Units of a Finite Field

The Group of Units of Zn

Splitting Fields

Concluding Activities

Exercises

Connections

Groups of Order 8 and 12: Semidirect Products of Groups

Introduction

Groups of Order 8

Semi-direct Products of Groups

Groups of Order 12 and p3

Concluding Activities

Exercises

Connections

Appendices

Functions

Special Types of Functions: Injections and Surjections

Composition of Functions

Inverse Functions

Theorems about Inverse Functions

Concluding Activities

Exercises

Mathematical Induction and the Well-Ordering Principle

Introduction

The Principle of Mathematical Induction

The Extended Principle of Mathematical Induction

The Strong Form of Mathematical Induction

TheWell-Ordering Principle

The Equivalence of the Well-Ordering Principle and the Principles of Mathematical Induction.

Concluding Activities

Exercises

About the Series

Textbooks in Mathematics

Learn more…

Subject Categories

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