1st Edition

# Abstract Algebra An Inquiry Based Approach

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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.

**The IntegersThe 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 SystemsOther 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

RingsRings

**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

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

**Introduction**

__Ideals and Homomorphisms__

More Ring Theory

More Ring Theory

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

**Introduction**

Symmetry

Symmetry

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

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

**Special Types of Functions: Injections and Surjections**

__Appendices__

FunctionsComposition 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

### Biography

Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom

"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 Chirteş,Zentralblatt MATH1295