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Abstract Algebra: An Integrated Approach
 
Joseph H. Silverman Brown University, Providence, RI
Abstract Algebra
Softcover ISBN:  978-1-4704-6860-6
Product Code:  AMSTEXT/55
List Price: $49.00
MAA Member Price: $44.10
AMS Member Price: $39.20
eBook ISBN:  978-1-4704-6902-3
Product Code:  AMSTEXT/55.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $36.00
Softcover ISBN:  978-1-4704-6860-6
eBook: ISBN:  978-1-4704-6902-3
Product Code:  AMSTEXT/55.B
List Price: $94.00 $71.50
MAA Member Price: $84.60 $64.35
AMS Member Price: $75.20 $57.20
Abstract Algebra
Click above image for expanded view
Abstract Algebra: An Integrated Approach
Joseph H. Silverman Brown University, Providence, RI
Softcover ISBN:  978-1-4704-6860-6
Product Code:  AMSTEXT/55
List Price: $49.00
MAA Member Price: $44.10
AMS Member Price: $39.20
eBook ISBN:  978-1-4704-6902-3
Product Code:  AMSTEXT/55.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $36.00
Softcover ISBN:  978-1-4704-6860-6
eBook ISBN:  978-1-4704-6902-3
Product Code:  AMSTEXT/55.B
List Price: $94.00 $71.50
MAA Member Price: $84.60 $64.35
AMS Member Price: $75.20 $57.20
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 552022; 567 pp
    MSC: Primary 12; 13; 16; 20

    This abstract algebra textbook takes an integrated approach that highlights the similarities of fundamental algebraic structures among a number of topics. The book begins by introducing groups, rings, vector spaces, and fields, emphasizing examples, definitions, homomorphisms, and proofs. The goal is to explain how all of the constructions fit into an axiomatic framework and to emphasize the importance of studying those maps that preserve the underlying algebraic structure. This fast-paced introduction is followed by chapters in which each of the four main topics is revisited and deeper results are proven.

    The second half of the book contains material of a more advanced nature. It includes a thorough development of Galois theory, a chapter on modules, and short surveys of additional algebraic topics designed to whet the reader's appetite for further study.

    This book is intended for a first introduction to abstract algebra and requires only a course in linear algebra as a prerequisite. The more advanced material could be used in an introductory graduate-level course.

    Ancillaries:

    Readership

    Undergraduate and graduate students interested in abstract algebra.

  • Table of Contents
     
     
    • Preface
    • Chapter 1. A Potpourri of Preliminary Topics
    • 1.1. What Are Definitions, Axioms, and Proofs?
    • 1.2. Mathematical Credos to Live By!
    • 1.3. A Smidgeon of Mathematical Logic and Some Proof Techniques
    • 1.4. A Smidgeon of Set Theory
    • 1.5. Functions
    • 1.6. Equivalence Relations
    • 1.7. Mathematical Induction
    • 1.8. A Smidgeon of Number Theory
    • 1.9. A Smidgeon of Combinatorics
    • Exercises
    • Chapter 2. Groups — Part 1
    • 2.1. Introduction to Groups
    • 2.2. Abstract Groups
    • 2.3. Interesting Examples of Groups
    • 2.4. Group Homomorphisms
    • 2.5. Subgroups, Cosets, and Lagrange's Theorem
    • 2.6. Products of Groups
    • Exercises
    • Chapter 3. Rings — Part 1
    • 3.1. Introduction to Rings
    • 3.2. Abstract Rings and Ring Homomorphisms
    • 3.3. Interesting Examples of Rings
    • 3.4. Some Important Special Types of Rings
    • 3.5. Unit Groups and Product Rings
    • 3.6. Ideals and Quotient Rings
    • 3.7. Prime Ideals and Maximal Ideals
    • Exercises
    • Chapter 4. Vector Spaces — Part 1
    • 4.1. Introduction to Vector Spaces
    • 4.2. Vector Spaces and Linear Transformations
    • 4.3. Interesting Examples of Vector Spaces
    • 4.4. Bases and Dimension
    • Exercises
    • Chapter 5. Fields — Part 1
    • 5.1. Introduction to Fields
    • 5.2. Abstract Fields and Homomorphisms
    • 5.3. Interesting Examples of Fields
    • 5.4. Subfields and Extension Fields
    • 5.5. Polynomial Rings
    • 5.6. Building Extension Fields
    • 5.7. Finite Fields
    • Exercises
    • Chapter 6. Groups — Part 2
    • 6.1. Normal Subgroups and Quotient Groups
    • 6.2. Groups Acting on Sets
    • 6.3. The Orbit-Stabilizer Counting Theorem
    • 6.4. Sylow's Theorem
    • 6.5. Two Counting Lemmas
    • 6.6. Double Cosets and Sylow's Theorem
    • Exercises
    • Chapter 7. Rings — Part 2
    • 7.1. Irreducible Elements and Unique Factorization Domains
    • 7.2. Euclidean Domains and Principal Ideal Domains
    • 7.3. Factorization in Principal Ideal Domains
    • 7.4. The Chinese Remainder Theorem
    • 7.5. Field of Fractions
    • 7.6. Multivariate and Symmetric Polynomials
    • Exercises
    • Chapter 8. Fields — Part 2
    • 8.1. Algebraic Numbers and Transcendental Numbers
    • 8.2. Polynomial Roots and Multiplicative Subgroups
    • 8.3. Splitting Fields, Separability, and Irreducibility
    • 8.4. Finite Fields Revisited
    • 8.5. Gauss's Lemma and Eisenstein's Irreducibility Criterion
    • 8.6. Ruler and Compass Constructions
    • Exercises
    • Chapter 9. Galois Theory: Fields+Groups
    • 9.1. What Is Galois Theory?
    • 9.2. A Quick Review of Polynomials and Field Extensions
    • 9.3. Fields of Algebraic Numbers
    • 9.4. Algebraically Closed Fields
    • 9.5. Automorphisms of Fields
    • 9.6. Splitting Fields — Part 1
    • 9.7. Splitting Fields — Part 2
    • 9.8. The Primitive Element Theorem
    • 9.9. Galois Extensions
    • 9.10. The Fundamental Theorem of Galois Theory
    • 9.11. Application: The Fundamental Theorem of Algebra
    • 9.12. Galois Theory of Finite Fields
    • 9.13. A Plethora of Galois Equivalences
    • 9.14. Cyclotomic Fields and Kummer Fields
    • 9.15. Application: Insolubility of Polynomial Equations by Radicals
    • 9.16. Linear Independence of Field Automorphisms
    • Exercises
    • Chapter 10. Vector Spaces — Part 2
    • 10.1. Vector Space Homomorphisms (aka Linear Transformations)
    • 10.2. Endomorphisms and Automorphisms
    • 10.3. Linear Transformations and Matrices
    • 10.4. Subspaces and Quotient Spaces
    • 10.5. Eigenvalues and Eigenvectors
    • 10.6. Determinants
    • 10.7. Determinants, Eigenvalues, and Characteristic Polynomials
    • 10.8. Inifinite-Dimensional Vector Spaces
    • Exercises
    • Chapter 11. Modules — Part 1:Rings+Vector-Like Spaces
    • 11.1. What Is a Module?
    • 11.2. Examples of Modules
    • 11.3. Submodules and Quotient Modules
    • 11.4. Free Modules and Finitely Generated Modules
    • 11.5. Homomorphisms, Endomorphisms, Matrices
    • 11.6. Noetherian Rings and Modules
    • 11.7. Matrices with Entries in a Euclidean Domain
    • 11.8. Finitely Generated Modules over Euclidean Domains
    • 11.9. Applications of the Structure Theorem
    • Exercises
    • Chapter 12. Groups — Part 3
    • 12.1. Permutation Groups
    • 12.2. Cayley's Theorem
    • 12.3. Simple Groups
    • 12.4. Composition Series
    • 12.5. Automorphism Groups
    • 12.6. Semidirect Products of Groups
    • 12.7. The Structure of Finite Abelian Groups
    • Exercises
    • Chapter 13. Modules — Part 2: Multilinear Algebra
    • 13.1. Multilinear Maps and Multilinear Forms
    • 13.2. Symmetric and Alternating Forms
    • 13.3. Alternating Forms on Free Modules
    • 13.4. The Determinant Map
    • Exercises
    • Chapter 14. Additional Topics in Brief
    • 14.1. Sets Countable and Uncountable
    • 14.2. The Axiom of Choice
    • 14.3. Tensor Products and Multilinear Algebra
    • 14.4. Commutative Algebra
    • 14.5. Category Theory
    • 14.6. Graph Theory
    • 14.7. Representation Theory
    • 14.8. Elliptic Curves
    • 14.9. Algebraic Number Theory
    • 14.10. Algebraic Geometry
    • 14.11. Euclidean Lattices
    • 14.12. Non-Commutative Rings
    • 14.13. Mathematical Cryptography
    • Exercises
    • Sample Syllabi
    • List of Notation
    • List of Figures
    • Index
  • Reviews
     
     
    • It will come as no surprise that the material is presented in a clear and flawless manner; in addition, there are many exercises and an extensive index.

      Franz Lemmermeyer, zbMATH Open
    • A quick review of these archives alone will show that textbooks for undergraduate abstract algebra courses are not in short supply. Several of them are excellent, and as an instructor, I have an embarrassment of riches in choosing for my course. I expect this text will be on that list when next I get to teach the subject. Silverman's dedication says, "This one is for the next generation." Indeed, this is a wonderful resource for training the next generation of mathematicians.

      Michele Intermont (Kalamazoo College), MAA Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 552022; 567 pp
MSC: Primary 12; 13; 16; 20

This abstract algebra textbook takes an integrated approach that highlights the similarities of fundamental algebraic structures among a number of topics. The book begins by introducing groups, rings, vector spaces, and fields, emphasizing examples, definitions, homomorphisms, and proofs. The goal is to explain how all of the constructions fit into an axiomatic framework and to emphasize the importance of studying those maps that preserve the underlying algebraic structure. This fast-paced introduction is followed by chapters in which each of the four main topics is revisited and deeper results are proven.

The second half of the book contains material of a more advanced nature. It includes a thorough development of Galois theory, a chapter on modules, and short surveys of additional algebraic topics designed to whet the reader's appetite for further study.

This book is intended for a first introduction to abstract algebra and requires only a course in linear algebra as a prerequisite. The more advanced material could be used in an introductory graduate-level course.

Ancillaries:

Readership

Undergraduate and graduate students interested in abstract algebra.

  • Preface
  • Chapter 1. A Potpourri of Preliminary Topics
  • 1.1. What Are Definitions, Axioms, and Proofs?
  • 1.2. Mathematical Credos to Live By!
  • 1.3. A Smidgeon of Mathematical Logic and Some Proof Techniques
  • 1.4. A Smidgeon of Set Theory
  • 1.5. Functions
  • 1.6. Equivalence Relations
  • 1.7. Mathematical Induction
  • 1.8. A Smidgeon of Number Theory
  • 1.9. A Smidgeon of Combinatorics
  • Exercises
  • Chapter 2. Groups — Part 1
  • 2.1. Introduction to Groups
  • 2.2. Abstract Groups
  • 2.3. Interesting Examples of Groups
  • 2.4. Group Homomorphisms
  • 2.5. Subgroups, Cosets, and Lagrange's Theorem
  • 2.6. Products of Groups
  • Exercises
  • Chapter 3. Rings — Part 1
  • 3.1. Introduction to Rings
  • 3.2. Abstract Rings and Ring Homomorphisms
  • 3.3. Interesting Examples of Rings
  • 3.4. Some Important Special Types of Rings
  • 3.5. Unit Groups and Product Rings
  • 3.6. Ideals and Quotient Rings
  • 3.7. Prime Ideals and Maximal Ideals
  • Exercises
  • Chapter 4. Vector Spaces — Part 1
  • 4.1. Introduction to Vector Spaces
  • 4.2. Vector Spaces and Linear Transformations
  • 4.3. Interesting Examples of Vector Spaces
  • 4.4. Bases and Dimension
  • Exercises
  • Chapter 5. Fields — Part 1
  • 5.1. Introduction to Fields
  • 5.2. Abstract Fields and Homomorphisms
  • 5.3. Interesting Examples of Fields
  • 5.4. Subfields and Extension Fields
  • 5.5. Polynomial Rings
  • 5.6. Building Extension Fields
  • 5.7. Finite Fields
  • Exercises
  • Chapter 6. Groups — Part 2
  • 6.1. Normal Subgroups and Quotient Groups
  • 6.2. Groups Acting on Sets
  • 6.3. The Orbit-Stabilizer Counting Theorem
  • 6.4. Sylow's Theorem
  • 6.5. Two Counting Lemmas
  • 6.6. Double Cosets and Sylow's Theorem
  • Exercises
  • Chapter 7. Rings — Part 2
  • 7.1. Irreducible Elements and Unique Factorization Domains
  • 7.2. Euclidean Domains and Principal Ideal Domains
  • 7.3. Factorization in Principal Ideal Domains
  • 7.4. The Chinese Remainder Theorem
  • 7.5. Field of Fractions
  • 7.6. Multivariate and Symmetric Polynomials
  • Exercises
  • Chapter 8. Fields — Part 2
  • 8.1. Algebraic Numbers and Transcendental Numbers
  • 8.2. Polynomial Roots and Multiplicative Subgroups
  • 8.3. Splitting Fields, Separability, and Irreducibility
  • 8.4. Finite Fields Revisited
  • 8.5. Gauss's Lemma and Eisenstein's Irreducibility Criterion
  • 8.6. Ruler and Compass Constructions
  • Exercises
  • Chapter 9. Galois Theory: Fields+Groups
  • 9.1. What Is Galois Theory?
  • 9.2. A Quick Review of Polynomials and Field Extensions
  • 9.3. Fields of Algebraic Numbers
  • 9.4. Algebraically Closed Fields
  • 9.5. Automorphisms of Fields
  • 9.6. Splitting Fields — Part 1
  • 9.7. Splitting Fields — Part 2
  • 9.8. The Primitive Element Theorem
  • 9.9. Galois Extensions
  • 9.10. The Fundamental Theorem of Galois Theory
  • 9.11. Application: The Fundamental Theorem of Algebra
  • 9.12. Galois Theory of Finite Fields
  • 9.13. A Plethora of Galois Equivalences
  • 9.14. Cyclotomic Fields and Kummer Fields
  • 9.15. Application: Insolubility of Polynomial Equations by Radicals
  • 9.16. Linear Independence of Field Automorphisms
  • Exercises
  • Chapter 10. Vector Spaces — Part 2
  • 10.1. Vector Space Homomorphisms (aka Linear Transformations)
  • 10.2. Endomorphisms and Automorphisms
  • 10.3. Linear Transformations and Matrices
  • 10.4. Subspaces and Quotient Spaces
  • 10.5. Eigenvalues and Eigenvectors
  • 10.6. Determinants
  • 10.7. Determinants, Eigenvalues, and Characteristic Polynomials
  • 10.8. Inifinite-Dimensional Vector Spaces
  • Exercises
  • Chapter 11. Modules — Part 1:Rings+Vector-Like Spaces
  • 11.1. What Is a Module?
  • 11.2. Examples of Modules
  • 11.3. Submodules and Quotient Modules
  • 11.4. Free Modules and Finitely Generated Modules
  • 11.5. Homomorphisms, Endomorphisms, Matrices
  • 11.6. Noetherian Rings and Modules
  • 11.7. Matrices with Entries in a Euclidean Domain
  • 11.8. Finitely Generated Modules over Euclidean Domains
  • 11.9. Applications of the Structure Theorem
  • Exercises
  • Chapter 12. Groups — Part 3
  • 12.1. Permutation Groups
  • 12.2. Cayley's Theorem
  • 12.3. Simple Groups
  • 12.4. Composition Series
  • 12.5. Automorphism Groups
  • 12.6. Semidirect Products of Groups
  • 12.7. The Structure of Finite Abelian Groups
  • Exercises
  • Chapter 13. Modules — Part 2: Multilinear Algebra
  • 13.1. Multilinear Maps and Multilinear Forms
  • 13.2. Symmetric and Alternating Forms
  • 13.3. Alternating Forms on Free Modules
  • 13.4. The Determinant Map
  • Exercises
  • Chapter 14. Additional Topics in Brief
  • 14.1. Sets Countable and Uncountable
  • 14.2. The Axiom of Choice
  • 14.3. Tensor Products and Multilinear Algebra
  • 14.4. Commutative Algebra
  • 14.5. Category Theory
  • 14.6. Graph Theory
  • 14.7. Representation Theory
  • 14.8. Elliptic Curves
  • 14.9. Algebraic Number Theory
  • 14.10. Algebraic Geometry
  • 14.11. Euclidean Lattices
  • 14.12. Non-Commutative Rings
  • 14.13. Mathematical Cryptography
  • Exercises
  • Sample Syllabi
  • List of Notation
  • List of Figures
  • Index
  • It will come as no surprise that the material is presented in a clear and flawless manner; in addition, there are many exercises and an extensive index.

    Franz Lemmermeyer, zbMATH Open
  • A quick review of these archives alone will show that textbooks for undergraduate abstract algebra courses are not in short supply. Several of them are excellent, and as an instructor, I have an embarrassment of riches in choosing for my course. I expect this text will be on that list when next I get to teach the subject. Silverman's dedication says, "This one is for the next generation." Indeed, this is a wonderful resource for training the next generation of mathematicians.

    Michele Intermont (Kalamazoo College), MAA Reviews
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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