Abstract Algebra by Dummit and Foote

In summary, "Abstract Algebra" by David Dummit and Richard Foote is a comprehensive textbook suitable for undergraduate students with a strong background in proofs and rigorous mathematics. It covers a wide range of topics in abstract algebra such as group theory, ring theory, modules, field theory, Galois theory, and representation theory. While the book is encyclopedic and has a wealth of examples, some readers may find it dry and tedious. It is recommended to supplement this book with other texts, such as Rotman's "Advanced Modern Algebra," for a better understanding and appreciation of the subject.

For those who have used this book


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Table of Contents:
Code:
[LIST]
[*] Preface
[*] Preliminaries
[LIST]
[*] Basics
[*] Properties of the Integers
[*] [itex]\mathbb{Z}/n\mathbb{Z}[/itex]: The Integers Modulo [itex]n[/itex]
[/LIST]
[*] Group Theory
[LIST]
[*] Introduction to Groups
[LIST]
[*] Basic Axioms and Examples
[*] Dihedral Groups
[*] Symmetric Groups
[*] Matrix Groups
[*] The Quaternion Group
[*] Homomorphisms and Isomorphisms 
[*] Group Actions
[/LIST]
[*] Subgroups
[LIST]
[*] Definition and Examples
[*] Centralizers and Normalizers, Stabilizers and Kernels
[*] Cyclic Groups and Cyclic Subgroups
[*] Subgroups Generated by Subsets of a Group
[*] The Lattice of Subgroups of a Group
[/LIST]
[*] Quotient Groups and Homomorphisms
[LIST]
[*] Definitions and Examples
[*] More on Cosets and Lagrange's Theorem
[*] The Isomorphism Theorems
[*] Composition Series and the Holder Program
[*] Transpositions and the Alternating Group
[/LIST]
[*] Group Actions
[LIST]
[*] Group Actions and Permutation Representations
[*] Groups Acting on Themselves by Left Multiplication—Cayley's Theorem
[*] Groups Acting on Themselves by Conjugation—The Class Equation
[*] Automorphisms
[*] The Sylow Theorems
[*] The Simplicity of [itex]A_n[/itex]
[/LIST]
[*] Direct and Semidirect Products and Abelian Groups
[LIST]
[*] Direct Products
[*] The Fundamental Theorem of Finitely Generated Abelian Groups
[*] Table of Groups of Small Order
[*] Recognizing Direct Products
[*] Semidirect Products
[/LIST]
[*] Further Topics in Group Theory
[LIST]
[*] [itex]p[/itex]-groups, Nilpotent Groups, and Solvable Groups
[*] Applications in Groups of Medium Order
[*] A Word on Free Groups
[/LIST]
[/LIST]
[*] Ring Theory
[LIST]
[*] Introduction to Rings
[LIST]
[*] Basic Definitions and Examples
[*] Examples: Polynomial Rings, Matrix Rings, and Group Rings
[*] Ring Homomorphisms an Quotient Rings
[*] Properties of Ideals
[*] Rings of Fractions
[*] The Chinese Remainder Theorem
[/LIST]
[*] Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains
[LIST]
[*] Euclidean Domains
[*] Principal Ideal Domains (P.I.D.s)
[*] Unique Factorization Domains (U.F.D.s)
[/LIST]
[*] Polynomial Rings
[LIST]
[*] Definitions and Basic Properties
[*] Polynomial Rings over Fields I
[*] Polynomial Rings that are Unique Factorization Domains
[*] Irreducibility Criteria
[*] Polynomial Rings over Fields II
[*] Polynomials in Several Variables over a Field and Grobner Bases
[/LIST]
[/LIST]
[*] Modules and Vector Spaces
[LIST]
[*] Introduction to Module Theory
[LIST]
[*] Basic Definitions and Examples
[*] Quotient Modules and Module Homomorphisms
[*] Generation of Modules, Direct Sums, and Free Modules
[*] Tensor Products of Modules
[*] Exact Sequences—Projective, Injective, and Flat Modules
[/LIST]
[*] Vector Spaces
[LIST]
[*] Definitions and Basic Theory
[*] The Matrix of a Linear Transformation
[*] Dual Vector Spaces
[*] Determinants
[*] Tensor Algebras, Symmetric and Exterior Algebras
[/LIST]
[*] Modules over Principal Ideal Domains
[LIST]
[*] The Basic Theory
[*] The Rational Canonical Form
[*] The Jordan Canonical Form
[/LIST]
[/LIST]
[*] Field Theory and Galois Theory
[LIST]
[*] Field Theory
[LIST]
[*] Basic Theory of Field Extensions
[*] Algebraic Extensions
[*] Classical Straightedge and Compass Constructions
[*] Splitting Fields and Algebraic Closures
[*] Separable and Inseparable Extensions
[*] Cyclotomic Polynomials and Extensions
[/LIST]
[*] Galois Theory
[LIST]
[*] Basic Definitions
[*] The Fundamental Theorem of Galois Theory
[*] Finite Fields
[*] Composite Extensions and Simple Extensions
[*] Cyclotomic Extensions and Abelian Extensions over [itex]\mathbb{Q}[/itex]
[*] Galois Groups of Polynomials
[*] Solvable and Radical Extensions: Insolvability of the Quintic
[*] Computation of Galois Groups over [itex]<\mathbb{Q}>[/itex]
[*] Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
[/LIST]
[/LIST]
[*] An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra
[LIST]
[*] Commutative Rings and Algebraic Geometry
[LIST]
[*] Noetherian Rings and Affine Algebraic Sets
[*] Radicals and Affine Varieties
[*] Integral Extensions and Hilbert's Nullstellensatz
[*] Localization
[*] The Prime Spectrum of a Ring
[/LIST]
[*] Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
[LIST]
[*] Artinian Rings
[*] Discrete Valuation Rings
[*] Dedekind Domains
[/LIST]
[*] Introduction to Homological Algebra and Group Cohomology
[LIST]
[*] Introduction to Homological Algebra—Ext and Tor
[*] The Cohomology of Groups
[*] Crossed Homomorphisms and [itex]H^1(G, A)[/itex]
[*] Group Extensions, Factor Sets and [itex]H^2(G, A)[/itex]
[/LIST]
[/LIST]
[*] Introduction to the Representation Theory of Finite Groups
[LIST]
[*] Representation Theory and Character Theory
[LIST]
[*] Linear Actions and Modules over Group Rings
[*] Wedderburn's Theorem and Some Consequences
[*] Character Theory and the Orthogonality Relations
[/LIST]
[*] Examples and Applications of Character Theory
[LIST]
[*] Characters of Groups of Small Order
[*] Theorems of Burnside and Hall
[*] Introduction to the Theory of Induced Characters
[/LIST]
[/LIST]
[*] Appendix: Cartesian Products and Zorn's Lemma
[*] Appendix: Category Theory
[*] Index
[/LIST]
 
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  • #2
I respect this book but I don't like reading it very much. It has great coverage of topics, but I find it very dry and tedious. Somehow it fails to convey any of the beauty or elegance of algebra. But it is encyclopedic and has a wealth of examples, so it is still well worth owning.

A much nicer book with similar scope is Rotman's Advanced Modern Algebra: https://www.amazon.com/dp/0821847414/?tag=pfamazon01-20 (be sure to get the 2nd edition, as it is a substantial improvement over the 1st). The exposition in this book is first-rate, and the proofs are generally much cleaner and less cluttered than those in Dummit and Foote.
 
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  • #3
i taught from this book and found it to have some very clear and memorable statements of results that help a student remember the main points. e.g. the statement of just when a group is a semi direct product is very clear and useful. and there were really a lot of good problems that greatly expanded the results too. but the proofs did not thrill me.

they sometimes omitted to check important details, or gave abstract proofs that were of no use in using the theorems. but algebraists much more accomplished than me in the field have used the book, e.g. Professor Parimala at Emory used it in her course, but i did not ask her opinion.
 
  • #4
I have mixed feelings about this book. It can be quite useful as a reference and for some explanations; that I cannot deny. On the other hand, sometimes this book just seems like too much of an encyclopedia to me. It doesn't get me excited about algebra.

I would think it ideal to start with Artin and then read Lang, supplementing Lang with examples and problems from Dummit/Foote, of which there are plenty (this is probably the book's main strength).
 
  • #5


Dear David Dummit and Richard Foote,

I am impressed by the comprehensive and rigorous approach taken in your book "Abstract Algebra". The clear organization of the content, from the basics of group theory to more advanced topics such as Galois theory and homological algebra, makes it a valuable resource for undergraduate students.

I appreciate the inclusion of prerequisites for readers, as well as the level of difficulty indicated in the title. This will help readers assess their readiness and choose the appropriate level of study.

Furthermore, the inclusion of examples and applications throughout the book makes the abstract concepts more tangible and relatable. The exercises at the end of each chapter also provide a valuable opportunity for students to apply their understanding and solidify their knowledge.

Overall, I highly recommend "Abstract Algebra" as a comprehensive and rigorous introduction to this important branch of mathematics. Thank you for your contribution to the field.

Sincerely,
 

FAQ: Abstract Algebra by Dummit and Foote

1. What is Abstract Algebra by Dummit and Foote?

Abstract Algebra by Dummit and Foote is a comprehensive textbook that covers the fundamental concepts of abstract algebra, including group theory, ring theory, and field theory. It is widely used as a textbook in undergraduate and graduate mathematics courses.

2. Who are the authors of Abstract Algebra by Dummit and Foote?

The authors of Abstract Algebra by Dummit and Foote are David S. Dummit and Richard M. Foote. They are both professors of mathematics at the University of Vermont and have extensive research and teaching experience in abstract algebra.

3. What makes Abstract Algebra by Dummit and Foote a popular textbook?

Abstract Algebra by Dummit and Foote is a popular textbook because it provides a thorough and rigorous treatment of abstract algebra topics while also being accessible to students with a basic understanding of algebra. It includes numerous examples, exercises, and applications that help students develop a strong understanding of the subject.

4. Is Abstract Algebra by Dummit and Foote suitable for self-study?

Yes, Abstract Algebra by Dummit and Foote is suitable for self-study. The textbook is well-organized and includes detailed explanations, examples, and exercises that make it easy for students to learn the material on their own. However, some prior knowledge of algebra and mathematical proof techniques may be helpful.

5. Are there any online resources available for Abstract Algebra by Dummit and Foote?

Yes, there are several online resources available for Abstract Algebra by Dummit and Foote. The authors have a website that includes additional exercises and solutions, as well as a list of errata for the textbook. There are also various study guides, lecture notes, and video lectures on abstract algebra that use Abstract Algebra by Dummit and Foote as a reference or textbook.

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