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Solution Manual for Abstract Algebra – 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
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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

Dummit And Foote Solution Manual

§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

$Manual$

§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

Solution Manual For Textbooks

§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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