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Author:David Dummit, Richard FooteTitle:Abstract AlgebraAmazon linkhttp://www.amazon.com/Abstract-Algebra-Edition-David-Dummit/dp/0471433349Prerequisities:Being acquainted with proofs and rigorous mathematics.Level:Undergrad

Table of Contents:

Code (Text):

[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

[/LIST]

[*] 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

[/LIST]

[*] 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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