Abstract Algebra : An Inquiry-Based Approach / Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom.

Format:

Book

Author/Creator:

Hodge, Jonathan K., 1980- author.

Schlicker, Steven, 1958- author.

Sundstrom, Theodore A., author.

Series:

Textbooks in mathematics (Boca Raton, Fla.)

Textbooks in Mathematics Series

Language:

English

Subjects (All):

Algebra, Abstract.

Physical Description:

1 online resource (546 pages)

Edition:

Second edition.

Place of Publication:

Boca Raton, FL : CRC Press, [2024]

Summary:

Abstract Algebra: An Inquiry-Based Approach, Second Edition not only teaches abstract algebra, but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The second edition of this unique, flexible approach builds on the success of the first edition. The authors offer an emphasis on active learning, helping students learn algebra by gradually building both their intuition and their ability to write coherent proofs in context. The goals for this text include: Allowing the flexibility to begin the course with either groups or rings. Introducing the ideas behind definitions and theorems to help students develop intuition. Helping students understand how mathematics is done. Students will experiment through examples, make conjectures, and then refine or prove their conjectures. Assisting students in developing their abilities to effectively communicate mathematical ideas. Actively involving students in realizing each of these goals through in-class and out-of-class activities, common in-class intellectual experiences, and challenging problem sets. Changes in the Second Edition Streamlining of introductory material with a quicker transition to the material on rings and groups. New investigations on extensions of fields and Galois theory. New exercises added and some sections reworked for clarity. More online Special Topics investigations and additional Appendices, including new appendices on other methods of proof and complex roots of unity. Encouraging students to do mathematics and be more than passive learners, this text shows students the way mathematics is developed is often different than how it is presented; definitions, theorems, and proofs do not simply appear fully formed; mathematical ideas are highly interconnected; and in abstract algebra, there is a considerable amount of intuition to be found.

Contents:

Cover

Half Title

Series Page

Title Page

Copyright Page

Contents

Note to Students

Preface

I. Number Systems

1. The Integers

Introduction

Arithmetic and Ordering Axioms

Divisibility in Z

Congruence

Factoring, Prime Numbers, and Greatest Common Divisors

Linear Combinations

Proofs of the Division Algorithm and the Fundamental Theorem of Arithmetic

Concluding Activities

Exercises

2. Equivalence Relations and Zn

Congruence Classes

Equivalence Relations

Equivalence Classes

The Number System Zn

Binary Operations

Zero Divisors and Units in Zn

3. Algebra in Other Number Systems

Subsets of the Real Numbers

The Complex Numbers

Matrices

Collections of Sets

Putting It All Together

II. Rings

4. Introduction to Rings

Basic Properties of Rings

Commutative Rings and Rings with Identity

Uniqueness of Identities and Inverses

Zero Divisors and Multiplicative Cancellation

Fields and Integral Domains

Connections

5. Integer Multiples and Exponents

Integer Multiplication and Exponentiation

Nonpositive Multiples and Exponents

Properties of Integer Multiplication and Exponentiation

The Characteristic of a Ring

6. Subrings, Extensions, and Direct Sums

The Subring Test

Subfields and Field Extensions

Direct Sums

7. Isomorphism and Invariants

Isomorphisms of Rings

Renaming Elements

Preserving Operations

Proving Isomorphism

Well-Defined Functions

Disproving Isomorphism

Invariants

Exercises.

Connections

III. Polynomial Rings

8. Polynomial Rings

Polynomials over Commutative Rings

Polynomials over an Integral Domain

Polynomial Functions

9. Divisibility in Polynomial Rings

The Division Algorithm in F[x]

Greatest Common Divisors of Polynomials

Relatively Prime Polynomials

The Euclidean Algorithm for Polynomials

10. Roots, Factors, and Irreducible Polynomials

Polynomial Functions and Remainders

Roots of Polynomials and the Factor Theorem

Irreducible Polynomials

Unique Factorization in F[x]

11. Irreducible Polynomials

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

12. Quotients of Polynomial Rings

Congruence Modulo a Polynomial

Congruence Classes of Polynomials

The Set F[x]/&lt

f(x)&gt

Special Quotients of Polynomial Rings

Algebraic Numbers

IV. More Ring Theory

13. Ideals and Homomorphisms

Ideals

Congruence Modulo an Ideal

Maximal and Prime Ideals

Homomorphisms

The Kernel and Image of a Homomorphism

The First Isomorphism Theorem for Rings

14. Divisibility and Factorization in Integral Domains

Divisibility and Euclidean Domains

Primes and Irreducibles

Unique Factorization Domains

Proof 1: Generalizing Greatest Common Divisors.

Proof 2: Principal Ideal Domains

15. From Z to C

From W to Z

Ordered Rings

From Z to Q

Ordering on Q

From Q to R

From R to C

A Characterization of the Integers

V. Groups

16. Symmetry

Symmetries

Symmetries of Regular Polygons

17. 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

18. Integer Powers of Elements in a Group

Powers of Elements in a Group

19. Subgroups

The Subgroup Test

The Center of a Group

The Subgroup Generated by an Element

20. Subgroups of Cyclic Groups

Subgroups of Cyclic Groups

Properties of the Order of an Element

Finite Cyclic Groups

Infinite Cyclic Groups

21. The Dihedral Groups

Relationships between Elements in Dn

Generators and Group Presentations

22. The Symmetric Groups

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

23. Cosets and Lagrange's Theorem

A Relation in Groups

Cosets

Lagrange's Theorem

Connections.

24. Normal Subgroups and Quotient Groups

An Operation on Cosets

Normal Subgroups

Quotient Groups

Cauchy's Theorem for Finite Abelian Groups

Simple Groups and the Simplicity of An

25. Products of Groups

External Direct Products of Groups

Orders of Elements in Direct Products

Internal Direct Products in Groups

26. Group Isomorphisms and Invariants

Isomorphisms of Groups

Some Basic Properties of Isomorphisms

Isomorphism Classes

Isomorphisms and Cyclic Groups

Cayley's Theorem

27. Homomorphisms and Isomorphism Theorems

The Kernel of a Homomorphism

The Image of a Homomorphism

The Isomorphism Theorems for Groups

The First Isomorphism Theorem for Groups

The Second Isomorphism Theorem for Groups

The Third Isomorphism Theorem for Groups

The Fourth Isomorphism Theorem for Groups

28. The Fundamental Theorem of Finite Abelian Groups

The Components: p-Groups

The Fundamental Theorem

29. The First Sylow Theorem

Conjugacy and the Class Equation

The Class Equation

Cauchy's Theorem

The First Sylow Theorem

The Second and Third Sylow Theorems

30. The Second and Third Sylow Theorems

Conjugate Subgroups and Normalizers

The Second Sylow Theorem

The Third Sylow Theorem

VI. Fields and Galois Theory

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

Finite Fields

The Group of Units of a Finite Field

The Group of Units of Zn

Splitting Fields

32. Extensions of Fields

A Quick Review of Linear Algebra

Extension Fields and the Degree of an Extension

Field Automorphisms

33. Galois Theory

The Galois Group

The Fundamental Theorem of Galois Theory

Solvability by Radicals

Solvable Groups

Polynomials Not Solvable By Radicals

Index.

Notes:

Includes index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Hodge, Jonathan K. Abstract Algebra

ISBN:

9781003814122

9781032634906

9781003814184

OCLC:

1410452320