Abstract Algebra by Dummit and Foote

[micromass]

• Author: David Dummit, Richard Foote
• Title: Abstract Algebra
• Amazon link https://www.amazon.com/dp/0471433349/?tag=pfamazon01-20
• Prerequisities: Being acquainted with proofs and rigorous mathematics.
• Level: Undergrad

Table of Contents:

[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]

 

[jbunniii]

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.

 

[mathwonk]

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.

 

[A. Bahat]

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).