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Abstract Algebra
An Inquiry Based Approach
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Book Description
To learn and understand mathematics, students must engage in the process of doing mathematics. Emphasizing active learning, Abstract Algebra: An InquiryBased 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 ringsfirst and groupsfirst 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.
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
TheWellOrdering 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
WellOrdering, 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
WellDefined 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: pGroups
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
Semidirect 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 WellOrdering Principle
Introduction
The Principle of Mathematical Induction
The Extended Principle of Mathematical Induction
The Strong Form of Mathematical Induction
TheWellOrdering Principle
The Equivalence of the WellOrdering Principle and the Principles of Mathematical Induction.
Concluding Activities
Exercises
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 Chirteş, Zentralblatt MATH 1295
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