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.
Author(s): Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom
Series: Textbooks in Mathematics
Edition: 2
Publisher: Chapman and Hall/CRC
Year: 2023
Language: English
Commentary: Publisher PDF | Published: December 19, 2023
Pages: 547
City: Boca Raton, FL
Tags: Algebra; Abstract Algebra; Integers; Rings; Ring Theory; Groups; Fields; Galois Theory
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
Concluding Activities
Exercises
3. Algebra in Other Number Systems
Introduction
Subsets of the Real Numbers
The Complex Numbers
Matrices
Collections of Sets
Putting It All Together
Concluding Activities
Exercises
II. Rings
4. 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
5. 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
6. Subrings, Extensions, and Direct Sums
Introduction
The Subring Test
Subfields and Field Extensions
Direct Sums
Concluding Activities
Exercises
Connections
7. Isomorphism and Invariants
Introduction
Isomorphisms of Rings
Renaming Elements
Preserving Operations
Proving Isomorphism
Well-Defined Functions
Disproving Isomorphism
Invariants
Concluding Activities
Exercises
Connections
III. Polynomial Rings
8. Polynomial Rings
Polynomials over Commutative Rings
Polynomials over an Integral Domain
Polynomial Functions
Concluding Activities
Exercises
Connections
9. 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
10. 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
11. 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
Concluding Activities
Exercises
Connections
12. Quotients of Polynomial Rings
Introduction
Congruence Modulo a Polynomial
Congruence Classes of Polynomials
The Set F[x]/
Special Quotients of Polynomial Rings
Algebraic Numbers
Concluding Activities
Exercises
Connections
IV. More Ring Theory
13. Ideals and Homomorphisms
Introduction
Ideals
Congruence Modulo an Ideal
Maximal and Prime Ideals
Homomorphisms
The Kernel and Image of a Homomorphism
The First Isomorphism Theorem for Rings
Concluding Activities
Exercises
Connections
14. 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
15. From Z to C
Introduction
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
Concluding Activities
Exercises
Connections
V. Groups
16. Symmetry
Introduction
Symmetries
Symmetries of Regular Polygons
Concluding Activities
Exercises
Connections
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
Concluding Activities
Exercises
Connections
18. Integer Powers of Elements in a Group
Introduction
Powers of Elements in a Group
Concluding Activities
Exercises
Connections
19. Subgroups
Introduction
The Subgroup Test
The Center of a Group
The Subgroup Generated by an Element
Concluding Activities
Exercises
Connections
20. 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
Connections
21. The Dihedral Groups
Introduction
Relationships between Elements in Dn
Generators and Group Presentations
Concluding Activities
Exercises
Connections
22. 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
23. Cosets and Lagrange’s Theorem
Introduction
A Relation in Groups
Cosets
Lagrange’s Theorem
Concluding Activities
Exercises
Connections
24. 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
25. Products of Groups
External Direct Products of Groups
Orders of Elements in Direct Products
Internal Direct Products in Groups
Concluding Activities
Exercises
Connections
26. Group Isomorphisms and Invariants
Introduction
Isomorphisms of Groups
Renaming Elements
Preserving Operations
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
27. Homomorphisms and Isomorphism Theorems
Homomorphisms
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
Concluding Activities
Exercises
Connections
28. The Fundamental Theorem of Finite Abelian Groups
Introduction
The Components: p-Groups
The Fundamental Theorem
Concluding Activities
Exercises
Connections
29. The First Sylow Theorem
Introduction
Conjugacy and the Class Equation
The Class Equation
Cauchy’s Theorem
The First Sylow Theorem
The Second and Third Sylow Theorems
Concluding Activities
Exercises
Connections
30. The Second and Third Sylow Theorems
Introduction
Conjugate Subgroups and Normalizers
The Second Sylow Theorem
The Third Sylow Theorem
Concluding Activities
Exercises
Connections
VI. Fields and Galois Theory
31. 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
32. Extensions of Fields
Introduction
A Quick Review of Linear Algebra
Extension Fields and the Degree of an Extension
Field Automorphisms
Concluding Activities
Exercises
Connections
33. Galois Theory
Introduction
The Galois Group
The Fundamental Theorem of Galois Theory
Solvability by Radicals
Solvable Groups
Polynomials Not Solvable By Radicals
Concluding Activities
Exercises
Connections
Index
