Softcover ISBN:  9781470468606 
Product Code:  AMSTEXT/55 
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AMS Member Price:  $39.20 
eBook ISBN:  9781470469023 
Product Code:  AMSTEXT/55.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $36.00 
Softcover ISBN:  9781470468606 
eBook: ISBN:  9781470469023 
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 
Softcover ISBN:  9781470468606 
Product Code:  AMSTEXT/55 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
eBook ISBN:  9781470469023 
Product Code:  AMSTEXT/55.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $36.00 
Softcover ISBN:  9781470468606 
eBook ISBN:  9781470469023 
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 DetailsPure and Applied Undergraduate TextsVolume: 55; 2022; 567 ppMSC: 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 fastpaced 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 graduatelevel course.
Ancillaries:ReadershipUndergraduate 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 OrbitStabilizer 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. InifiniteDimensional Vector Spaces

Exercises

Chapter 11. Modules — Part 1:Rings+VectorLike 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. NonCommutative Rings

14.13. Mathematical Cryptography

Exercises

Sample Syllabi

List of Notation

List of Figures

Index


Additional Material

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


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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 fastpaced 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 graduatelevel course.
Ancillaries:
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 OrbitStabilizer 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. InifiniteDimensional Vector Spaces

Exercises

Chapter 11. Modules — Part 1:Rings+VectorLike 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. NonCommutative 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