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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

Chapter 2: Subgroups

Chapter 3: Quotient Groups and Homomorphisms

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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  • §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
    (#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
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  • §5.3: Table of Groups of Small Order
    (#1)
  • §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

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
    (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11)

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
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  • §9.3: Polynomial Rings that are Unique Factorization Domains
    (#1) (#2) (#3) (#4) (#5)
  • §9.4: Irreducibility Criteria
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  • §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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  • §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
    (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8)
  • §13.2: Algebraic Extensions
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  • §13.3: Classical Straightedge and Compass Constructions
    (#1) (#2) (#3) (#4) (#5)
  • §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
    (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)
  • §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
    (#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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  • §14.7: Solvable and Radical Extensions: Insolvability of the Quintic
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  • §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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  • §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
    (#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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  • §17.3: Crossed Homomorphisms and H^1(G,A)
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  • §17.4: Group Extensions, Factor Sets, and H^2(G,A)
    (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10)

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
    (#1) (#2) (#3) (#4) (#5)
  • §19.3: Introduction to the Theory of Induced Characters
    (#1) (#2) (#3) (#4) (#5) (#6) (#7) (#8) (#9) (#10) (#11) (#12) (#13) (#14) (#15)

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