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# Algebra Abstract Algebra by Dummit and Foote

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 Mentor P: 18,040 Author: David Dummit, Richard Foote Title: Abstract Algebra Amazon link http://www.amazon.com/Abstract-Algeb.../dp/0471433349 Prerequisities: Being acquainted with proofs and rigorous mathematics. Level: Undergrad Table of Contents:  Preface Preliminaries Basics Properties of the Integers $\mathbb{Z}/n\mathbb{Z}$: The Integers Modulo $n$ Group Theory Introduction to Groups Basic Axioms and Examples Dihedral Groups Symmetric Groups Matrix Groups The Quaternion Group Homomorphisms and Isomorphisms Group Actions Subgroups 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 Quotient Groups and Homomorphisms Definitions and Examples More on Cosets and Lagrange's Theorem The Isomorphism Theorems Composition Series and the Holder Program Transpositions and the Alternating Group Group Actions 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 $A_n$ Direct and Semidirect Products and Abelian Groups Direct Products The Fundamental Theorem of Finitely Generated Abelian Groups Table of Groups of Small Order Recognizing Direct Products Semidirect Products Further Topics in Group Theory $p$-groups, Nilpotent Groups, and Solvable Groups Applications in Groups of Medium Order A Word on Free Groups Ring Theory Introduction to Rings 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 Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains Euclidean Domains Principal Ideal Domains (P.I.D.s) Unique Factorization Domains (U.F.D.s) Polynomial Rings 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 Modules and Vector Spaces Introduction to Module Theory 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 Vector Spaces Definitions and Basic Theory The Matrix of a Linear Transformation Dual Vector Spaces Determinants Tensor Algebras, Symmetric and Exterior Algebras Modules over Principal Ideal Domains The Basic Theory The Rational Canonical Form The Jordan Canonical Form Field Theory and Galois Theory Field Theory 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 Galois Theory Basic Definitions The Fundamental Theorem of Galois Theory Finite Fields Composite Extensions and Simple Extensions Cyclotomic Extensions and Abelian Extensions over $\mathbb{Q}$ Galois Groups of Polynomials Solvable and Radical Extensions: Insolvability of the Quintic Computation of Galois Groups over $<\mathbb{Q}>$ Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra Commutative Rings and Algebraic Geometry Noetherian Rings and Affine Algebraic Sets Radicals and Affine Varieties Integral Extensions and Hilbert's Nullstellensatz Localization The Prime Spectrum of a Ring Artinian Rings, Discrete Valuation Rings, and Dedekind Domains Artinian Rings Discrete Valuation Rings Dedekind Domains Introduction to Homological Algebra and Group Cohomology Introduction to Homological Algebra—Ext and Tor The Cohomology of Groups Crossed Homomorphisms and $H^1(G, A)$ Group Extensions, Factor Sets and $H^2(G, A)$ Introduction to the Representation Theory of Finite Groups Representation Theory and Character Theory Linear Actions and Modules over Group Rings Wedderburn's Theorem and Some Consequences Character Theory and the Orthogonality Relations Examples and Applications of Character Theory Characters of Groups of Small Order Theorems of Burnside and Hall Introduction to the Theory of Induced Characters Appendix: Cartesian Products and Zorn's Lemma Appendix: Category Theory Index