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Solution to Abstract Algebra by Dummit & Foote 3rd edition
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Chapter 0: Preliminaries
§0.1: Basics
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§0.2: Properties of the Integers
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§0.3: Z/(n) – The Integers Modulo n
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Chapter 1: Introduction to Groups
§1.1: Basic Axioms and Examples
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§1.2: Dihedral Groups
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§1.3: Symmetric Groups
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§1.4: Matrix Groups
§1.5: The Quaternion Group
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§1.6: Homomorphisms and Isomorphisms
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§1.7: Group Actions
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Chapter 2: Subgroups
§2.1: Definition and Examples
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§2.2: Centralizers and Normalizers, Stabilizers and Kernels
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§2.3: Cyclic Groups and Cyclic Subgroups
§2.4: Subgroups Generated by Subsets of a Group
§2.5: The Lattice of Subgroups of a Group
Chapter 3: Quotient Groups and Homomorphisms
§3.1: Definitions and Examples
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§3.2: More on Cosets and Lagrange's Theorem
§3.3: The Isomorphism Theorems
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§3.4: Composition Series and the Hölder Program
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§3.5: Transpositions and the Alternating Group
Chapter 4: Group Actions
§4.1: Group Actions and Permutation Representations
§4.2: Groups Acting on Themselves by Left Multiplication – Cayley's Theorem
§4.3: Groups Acting on Themselves by Conjugation – The Class Equation
§4.4: Automorphisms
§4.5: The Sylow Theorems
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§4.6: The Simplicity of A_n
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Chapter 5: Direct and Semidirect Products and Abelian Groups
§5.1: Direct Products
§5.2: The Fundamental Theorem of Finitely Generated Abelian Groups
§5.3: Table of Groups of Small Order
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§5.4: Recognizing Direct Products
§5.5: Semidirect Products
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Chapter 6: Further Topics in Group Theory
§6.1: p-Groups, Nilpotent Groups, and Solvable Groups
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§6.2: Applications in Groups of Medium Order
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§6.3: A Word on Free Groups
Chapter 7: Introduction to Rings
§7.1: Basic Definitions and Examples
§7.2: Examples: Polynomial Rings, Matrix Rings, and Group Rings
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§7.3: Ring Homomorphisms and Quotient Rings
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§7.4: Properties of Ideals
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§7.5: Rings of Fractions
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§7.6: The Chinese Remainder Theorem
Chapter 8: Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains
§8.1: Euclidean Domains
§8.2: Principal Ideal Domains
§8.3: Unique Factorization Domains
Chapter 9: Polynomial Rings
§9.1: Definitions and Basic Properties
§9.2: Polynomial Rings over Fields I
§9.3: Polynomial Rings that are Unique Factorization Domains
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§9.4: Irreducibility Criteria
§9.5: Polynomial Rings over Fields II
§9.6: Polynomials in Several Variables over a Field and Gröbner Bases
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Chapter 10: Introduction to Module Theory
§10.1: Basic Definitions and Examples
§10.2: Quotient Modules and Module Homomorphisms
§10.3: Generation of Modules, Direct Sums, and Free Modules
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§10.4: Tensor Products of Modules
§10.5: Exact Sequences – Projective, Injective, and Flat Modules
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Chapter 11: Vector Spaces
§11.1: Definitions and Basic Theory
§11.2: The Matrix of a Linear Transformation
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§11.3: Dual Vector Spaces
§11.4: Determinants
§11.5: Tensor Algebras, Symmetric and Exterior Algebras
Chapter 12: Modules over Principal Ideal Domains
§12.1: The Basic Theory
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§12.2: The Rational Canonical Form
§12.3: The Jordan Canonical Form
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Chapter 13: Field Theory
§13.1: Basic Theory of Field Extensions
§13.2: Algebraic Extensions
§13.3: Classical Straightedge and Compass Constructions
§13.4: Splitting Fields and Algebraic Closures
§13.5: Separable and Inseparabel Extensions
§13.6: Cyclotomic Polynomials and Extensions
Chapter 14: Galois Theory
§14.1: Basic Definitions
§14.2: The Fundamental Theorem of Galois Theory
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§14.3: Finite Fields
§14.4: Composite Extensions and Simple Extensions
§14.5: Cyclotomic Extensions and Abelian Extensions over \mathbb{Q}
§14.6: Galois Groups of Polynomials
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§14.7: Solvable and Radical Extensions: Insolvability of the Quintic
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§14.8: Computation of Galois Groups over \mathbb{Q}
§14.9: Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
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Chapter 15: Commutative Rings and Algebraic Geometry
§15.1: Noetherian Rings and Affine Algebraic Sets
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§15.2: Radicals and Affine Varieties
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§15.3: Integral Extensions and Hilbert's Nullstellensatz
§15.4: Localization
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§15.5: The Prime Spectrum of a Ring
Chapter 16: Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
§16.1: Artinian Rings
§16.2: Discrete Valuation Rings
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§16.3: Dedekind Domains
Chapter 17: Introduction to Homological Algebra and Group Cohomology
§17.1: Introduction to Homological Algebra – Ext and Tor
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§17.2: The Cohomology of Groups
§17.3: Crossed Homomorphisms and H^1(G,A)
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§17.4: Group Extensions, Factor Sets, and H^2(G,A)
Chapter 18: Representation Theory and Character Theory
§18.1: Linear Actions and Modules over Group Rings
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§18.2: Wedderburn's Theorem and Some Consequences
§18.3: Character Theory and the Orthogonality Relations
Chapter 19: Examples and Applications of Character Theory
§19.1: Characters of Groups of Small Order
§19.2: Theorems of Burnside and Hall
§19.3: Introduction to the Theory of Induced Characters
The set of prime ideals of a commutative ring contains inclusion-minimal elements
Use Zorn's Lemma to construct an ideal which maximally does not contain a given finitely generated ideal
Not every ideal is prime
Characterization of maximal ideals in the ring of all continuous real-valued functions
Definition and basic properties of the Jacobson radical of an ideal
In a commutative ring, prime ideals are radical
Definition and basic properties of the radical of an ideal
Prove that the augmentation ideal of a given group ring is nilpotent
An ideal which is finitely generated by nilpotent elements is nilpotent
Constructing units from nilpotent elements in a commutative ring
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