1st Edition

Abstract Algebra An Inquiry Based Approach

    596 Pages 31 B/W Illustrations
    by Chapman & Hall

    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 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

    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 MATH 1295