# Solution to Abstract Algebra by Dummit & Foote 3rd edition

We would like provide a complete solution manual to the book Abstract Algebra by Dummit & Foote 3rd edition. It will be updated regularly. Please also make a comment if you would like some particular problem to be updated.

### Chapter 0: Preliminaries

- §0.1: Basics

(#1) (#2) (#3) (#4) (#5) (#6) (#7) - §0.2: Properties of the Integers

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) - §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) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §1.3: Symmetric Groups

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §1.4: Matrix Groups

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) - §1.5: The Quaternion Group

(#1) (#2) (#3) - §1.6: Homomorphisms and Isomorphisms

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### Chapter 2: Subgroups

- §2.1: Definition and Examples

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) - §2.2: Centralizers and Normalizers, Stabilizers and Kernels

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §2.3: Cyclic Groups and Cyclic Subgroups

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) - §2.4: Subgroups Generated by Subsets of a Group

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §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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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) - §3.4: Composition Series and the Hölder Program

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) - §3.5: Transpositions and the Alternating Group

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### Chapter 4: Group Actions

- §4.1: Group Actions and Permutation Representations

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) - §4.2: Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §4.3: Groups Acting on Themselves by Conjugation – The Class Equation

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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §4.5: The Sylow Theorems

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### Chapter 5: Direct and Semidirect Products and Abelian Groups

- §5.1: Direct Products

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §5.2: The Fundamental Theorem of Finitely Generated Abelian Groups

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) - §5.3: Table of Groups of Small Order

(#1) - §5.4: Recognizing Direct Products

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §5.5: Semidirect Products

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### Chapter 6: Further Topics in Group Theory

- §6.1: p-Groups, Nilpotent Groups, and Solvable Groups

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) - §6.2: Applications in Groups of Medium Order

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### Chapter 7: Introduction to Rings

- §7.1: Basic Definitions and Examples

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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) - §7.3: Ring Homomorphisms and Quotient Rings

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(#1) (#2) (#3) (#4) (#5) (#6) - §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

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) - §8.2: Principal Ideal Domains

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - §8.3: Unique Factorization Domains

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### Chapter 9: Polynomial Rings

- §9.1: Definitions and Basic Properties

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §9.2: Polynomial Rings over Fields I

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) - §9.3: Polynomial Rings that are Unique Factorization Domains

(#1) (#2) (#3) (#4) (#5) - §9.4: Irreducibility Criteria

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) - §9.5: Polynomial Rings over Fields II

(#1) (#2) (#3) (#4) (#5) (#6) (#7) - §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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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §10.3: Generation of Modules, Direct Sums, and Free Modules

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### Chapter 11: Vector Spaces

- §11.1: Definitions and Basic Theory

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §11.2: The Matrix of a Linear Transformation

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(#1) (#2) (#3) (#4) (#5) - §11.4: Determinants

(#1) (#2) (#3) (#4) (#5) (#6) - §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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### Chapter 13: Field Theory

- §13.1: Basic Theory of Field Extensions

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - §13.2: Algebraic Extensions

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(#1) (#2) (#3) (#4) (#5) - §13.4: Splitting Fields and Algebraic Closures

(#1) (#2) (#3) (#4) (#5) (#6) - §13.5: Separable and Inseparabel Extensions

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) - §13.6: Cyclotomic Polynomials and Extensions

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### Chapter 14: Galois Theory

- §14.1: Basic Definitions

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) - §14.2: The Fundamental Theorem of Galois Theory

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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) - §14.4: Composite Extensions and Simple Extensions

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) - §14.5: Cyclotomic Extensions and Abelian Extensions over \mathbb{Q}

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §14.6: Galois Groups of Polynomials

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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) - §14.8: Computation of Galois Groups over \mathbb{Q}

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) - §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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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) (#36) (#37) (#38) (#39) (#40) - §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

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) - §16.2: Discrete Valuation Rings

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) - §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

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) (#19) (#20) (#21) (#22) (#23) (#24) (#25) (#26) (#27) (#28) (#29) (#30) (#31) (#32) (#33) (#34) (#35) - §17.2: The Cohomology of Groups

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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) - §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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(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) (#18) - §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

(#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15) (#16) (#17) - §19.2: Theorems of Burnside and Hall

(#1) (#2) (#3) (#4) (#5) - §19.3: Introduction to the Theory of Induced Characters

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