- #1
- 22,183
- 3,324
- Author: David Dummit, Richard Foote
- Title: Abstract Algebra
- Amazon link https://www.amazon.com/dp/0471433349/?tag=pfamazon01-20
- Prerequisities: Being acquainted with proofs and rigorous mathematics.
- Level: Undergrad
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Preliminaries
[LIST]
[*] Basics
[*] Properties of the Integers
[*] [itex]\mathbb{Z}/n\mathbb{Z}[/itex]: The Integers Modulo [itex]n[/itex]
[/LIST]
[*] Group Theory
[LIST]
[*] Introduction to Groups
[LIST]
[*] Basic Axioms and Examples
[*] Dihedral Groups
[*] Symmetric Groups
[*] Matrix Groups
[*] The Quaternion Group
[*] Homomorphisms and Isomorphisms
[*] Group Actions
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[*] Subgroups
[LIST]
[*] Definition and Examples
[*] Centralizers and Normalizers, Stabilizers and Kernels
[*] Cyclic Groups and Cyclic Subgroups
[*] Subgroups Generated by Subsets of a Group
[*] The Lattice of Subgroups of a Group
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[*] Quotient Groups and Homomorphisms
[LIST]
[*] Definitions and Examples
[*] More on Cosets and Lagrange's Theorem
[*] The Isomorphism Theorems
[*] Composition Series and the Holder Program
[*] Transpositions and the Alternating Group
[/LIST]
[*] Group Actions
[LIST]
[*] Group Actions and Permutation Representations
[*] Groups Acting on Themselves by Left Multiplication—Cayley's Theorem
[*] Groups Acting on Themselves by Conjugation—The Class Equation
[*] Automorphisms
[*] The Sylow Theorems
[*] The Simplicity of [itex]A_n[/itex]
[/LIST]
[*] Direct and Semidirect Products and Abelian Groups
[LIST]
[*] Direct Products
[*] The Fundamental Theorem of Finitely Generated Abelian Groups
[*] Table of Groups of Small Order
[*] Recognizing Direct Products
[*] Semidirect Products
[/LIST]
[*] Further Topics in Group Theory
[LIST]
[*] [itex]p[/itex]-groups, Nilpotent Groups, and Solvable Groups
[*] Applications in Groups of Medium Order
[*] A Word on Free Groups
[/LIST]
[/LIST]
[*] Ring Theory
[LIST]
[*] Introduction to Rings
[LIST]
[*] Basic Definitions and Examples
[*] Examples: Polynomial Rings, Matrix Rings, and Group Rings
[*] Ring Homomorphisms an Quotient Rings
[*] Properties of Ideals
[*] Rings of Fractions
[*] The Chinese Remainder Theorem
[/LIST]
[*] Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains
[LIST]
[*] Euclidean Domains
[*] Principal Ideal Domains (P.I.D.s)
[*] Unique Factorization Domains (U.F.D.s)
[/LIST]
[*] Polynomial Rings
[LIST]
[*] Definitions and Basic Properties
[*] Polynomial Rings over Fields I
[*] Polynomial Rings that are Unique Factorization Domains
[*] Irreducibility Criteria
[*] Polynomial Rings over Fields II
[*] Polynomials in Several Variables over a Field and Grobner Bases
[/LIST]
[/LIST]
[*] Modules and Vector Spaces
[LIST]
[*] Introduction to Module Theory
[LIST]
[*] Basic Definitions and Examples
[*] Quotient Modules and Module Homomorphisms
[*] Generation of Modules, Direct Sums, and Free Modules
[*] Tensor Products of Modules
[*] Exact Sequences—Projective, Injective, and Flat Modules
[/LIST]
[*] Vector Spaces
[LIST]
[*] Definitions and Basic Theory
[*] The Matrix of a Linear Transformation
[*] Dual Vector Spaces
[*] Determinants
[*] Tensor Algebras, Symmetric and Exterior Algebras
[/LIST]
[*] Modules over Principal Ideal Domains
[LIST]
[*] The Basic Theory
[*] The Rational Canonical Form
[*] The Jordan Canonical Form
[/LIST]
[/LIST]
[*] Field Theory and Galois Theory
[LIST]
[*] Field Theory
[LIST]
[*] Basic Theory of Field Extensions
[*] Algebraic Extensions
[*] Classical Straightedge and Compass Constructions
[*] Splitting Fields and Algebraic Closures
[*] Separable and Inseparable Extensions
[*] Cyclotomic Polynomials and Extensions
[/LIST]
[*] Galois Theory
[LIST]
[*] Basic Definitions
[*] The Fundamental Theorem of Galois Theory
[*] Finite Fields
[*] Composite Extensions and Simple Extensions
[*] Cyclotomic Extensions and Abelian Extensions over [itex]\mathbb{Q}[/itex]
[*] Galois Groups of Polynomials
[*] Solvable and Radical Extensions: Insolvability of the Quintic
[*] Computation of Galois Groups over [itex]<\mathbb{Q}>[/itex]
[*] Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
[/LIST]
[/LIST]
[*] An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra
[LIST]
[*] Commutative Rings and Algebraic Geometry
[LIST]
[*] Noetherian Rings and Affine Algebraic Sets
[*] Radicals and Affine Varieties
[*] Integral Extensions and Hilbert's Nullstellensatz
[*] Localization
[*] The Prime Spectrum of a Ring
[/LIST]
[*] Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
[LIST]
[*] Artinian Rings
[*] Discrete Valuation Rings
[*] Dedekind Domains
[/LIST]
[*] Introduction to Homological Algebra and Group Cohomology
[LIST]
[*] Introduction to Homological Algebra—Ext and Tor
[*] The Cohomology of Groups
[*] Crossed Homomorphisms and [itex]H^1(G, A)[/itex]
[*] Group Extensions, Factor Sets and [itex]H^2(G, A)[/itex]
[/LIST]
[/LIST]
[*] Introduction to the Representation Theory of Finite Groups
[LIST]
[*] Representation Theory and Character Theory
[LIST]
[*] Linear Actions and Modules over Group Rings
[*] Wedderburn's Theorem and Some Consequences
[*] Character Theory and the Orthogonality Relations
[/LIST]
[*] Examples and Applications of Character Theory
[LIST]
[*] Characters of Groups of Small Order
[*] Theorems of Burnside and Hall
[*] Introduction to the Theory of Induced Characters
[/LIST]
[/LIST]
[*] Appendix: Cartesian Products and Zorn's Lemma
[*] Appendix: Category Theory
[*] Index
[/LIST]
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