Abstract Algebra: David S. Dummit and Richard M. Foote

Solutions to exercises from Abstract Algebra by Dummit & Foote. (Not all links are active.)
Chapter 0: Preliminaries
Chapter 1: Introduction to Groups
Chapter 2: Subgroups
Chapter 3: Quotient Groups and Homomorphisms
Chapter 4: Group Actions
Chapter 5: Direct and Semidirect Products and Abelian Groups
Chapter 6: Further Topics in Group Theory
Chapter 7: Introduction to Rings
Chapter 8: Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains
Chapter 9: Polynomial Rings
Chapter 10: Introduction to Module Theory
Chapter 11: Vector Spaces
Chapter 12: Modules over Principal Ideal Domains
Chapter 13: Field Theory
Chapter 14: Galois Theory
Chapter 15: Commutative Rings and Algebraic Geometry
Chapter 16: Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
Chapter 17: Introduction to Homological Algebra and Group Cohomology
Chapter 18: Representation Theory and Character Theory
Chapter 19: Examples and Applications of Character Theory
Appendix I: Cartesian Products and Zorn’s Lemma
Appendix II: Category Theory

Solutions

  • 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
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    §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
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    §2.4: Subgroups Generated by Subsets of a Group
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    §2.5: The Lattice of Subgroups of a Group
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  • Chapter 3: Quotient Groups and Homomorphisms
    §3.1: Definitions and Examples
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    §3.2: More on Cosets and Lagrange’s Theorem
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    §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
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  • Chapter 4: Group Actions
    §4.1: Group Actions and Permutation Representations
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    §4.2: Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem
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    §4.3: Groups Acting on Themselves by Conjugation – The Class Equation
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    §4.4: Automorphisms
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    §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
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    §5.2: The Fundamental Theorem of Finitely Generated Abelian Groups
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    §5.3: Table of Groups of Small Order
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    §5.4: Recognizing Direct Products
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    §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
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  • Chapter 7: Introduction to Rings
    §7.1: Basic Definitions and Examples
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    §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
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  • Chapter 8: Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains
    §8.1: Euclidean Domains
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    §8.2: Principal Ideal Domains
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    §8.3: Unique Factorization Domains
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  • Chapter 9: Polynomial Rings
    §9.1: Definitions and Basic Properties
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    §9.2: Polynomial Rings over Fields I
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    §9.3: Polynomial Rings that are Unique Factorization Domains
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    §9.4: Irreducibility Criteria
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    §9.5: Polynomial Rings over Fields II
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    §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
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    §10.2: Quotient Modules and Module Homomorphisms
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    §10.3: Generation of Modules, Direct Sums, and Free Modules
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    §10.4: Tensor Products of Modules
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    §10.5: Exact Sequences – Projective, Injective, and Flat Modules
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  • Chapter 11: Vector Spaces
    §11.1: Definitions and Basic Theory
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    §11.2: The Matrix of a Linear Transformation
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    §11.3: Dual Vector Spaces
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    §11.4: Determinants
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    §11.5: Tensor Algebras, Symmetric and Exterior Algebras
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  • Chapter 12: Modules over Principal Ideal Domains
    §12.1: The Basic Theory
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    §12.2: The Rational Canonical Form
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    §12.3: The Jordan Canonical Form
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  • Chapter 13: Field Theory
    §13.1: Basic Theory of Field Extensions
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    §13.2: Algebraic Extensions
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    §13.3: Classical Straightedge and Compass Constructions
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    §13.4: Splitting Fields and Algebraic Closures
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    §13.5: Separable and Inseparabel Extensions
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    §13.6: Cyclotomic Polynomials and Extensions
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  • Chapter 14: Galois Theory
    §14.1: Basic Definitions
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    §14.2: The Fundamental Theorem of Galois Theory
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    §14.3: Finite Fields
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    §14.4: Composite Extensions and Simple Extensions
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    §14.5: Cyclotomic Extensions and Abelian Extensions over \mathbb{Q}
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    §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}
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    §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
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    §15.4: Localization
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    §15.5: The Prime Spectrum of a Ring
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  • Chapter 16: Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
    §16.1: Artinian Rings
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    §16.2: Discrete Valuation Rings
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    §16.3: Dedekind Domains
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  • 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
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    §17.3: Crossed Homomorphisms and H^1(G,A)
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    §17.4: Group Extensions, Factor Sets, and H^2(G,A)
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  • 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
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    §18.3: Character Theory and the Orthogonality Relations
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  • Chapter 19: Examples and Applications of Character Theory
    §19.1: Characters of Groups of Small Order
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    §19.2: Theorems of Burnside and Hall
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    §19.3: Introduction to the Theory of Induced Characters
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  • Appendix I: Cartesian Products and Zorn’s Lemma
  • Appendix II: Category Theory

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