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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 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 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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