Basic Abstract Algebra · for Graduate Students and Advanced Undergraduates by Ash, Robert B. -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Contents Preface Remarks on Expository Writing in Mathematics Chapter 0 Prerequisites

0.1 Elementary Number Theory 0.2 Set Theory 0.3 Linear Algebra

Chapter 1 Group Fundamentals

1.1 Groups and Subgroups 1.2 Permutation Groups 1.3 Cosets, Normal Subgroups and Homomorphisms 1.4 The Isomorphism Theorems 1.5 Direct Products

Chapter 2 Ring Fundamentals

2.1 Basic Definitions and Properties 2.2 Ideals, Homomorphisms and Quotient Rings 2.3 The Isomorphism Theorems For Rings 2.4 Maximal and Prime Ideals 2.5 Polynomial Rings 2.6 Unique Factorization 2.7 Principal Ideal Domains and Euclidean Domains 2.8 Rings of Fractions 2.9 Irreducible Polynomials

Chapter 3 Field Fundamentals

3.1 Field Extensions 3.2 Splitting Fields 3.3 Algebraic Closures 3.4 Separability 3.5 Normal Extensions

Chapter 4 Module Fundamentals

4.1 Modules and Algebras 4.2 The Isomorphism Theorems For Modules 4.3 Direct Sums and Free Modules 4.4 Homomorphisms and Matrices 4.5 Smith Normal Form 4.6 Fundamental Structure Theorems 4.7 Exact Sequences and Diagram Chasing

Enrichment

(Chapters 1–4)

Chapter 5 Some Basic Techniques of Group Theory

5.1 Groups Acting on Sets 5.2 The Orbit-Stabilizer Theorem 5.3 Applications to Combinatorics 5.4 The Sylow Theorems 5.5 Applications of the Sylow Theorems 5.6 Composition Series 5.7 Solvable and Nilpotent Groups 5.8 Generators and Relations

Chapter 6 Galois Theory

6.1 Fixed Fields and Galois Groups 6.2 The Fundamental Theorem 6.3 Computing a Galois Group Directly 6.4 Finite Fields 6.5 Cyclotomic Fields 6.6 The Galois Group of a Cubic 6.7 Cyclic and Kummer Extensions 6.8 Solvability by Radicals 6.9 Transcendental Extensions Appendix to Chapter 6

Chapter 7 Introducing Algebraic Number Theory

(Commutative Algebra 1) 7.1 Integral Extensions 7.2 Quadratic Extensions of the Rationals 7.3 Norms and Traces 7.4 The Discriminant 7.5 Noetherian and Artinian Modules and Rings 7.6 Fractional Ideals 7.7 Unique Factorization of Ideals in a Dedekind Domain 7.8 Some Arithmetic in Dedekind Domains 7.9 p-adic Numbers

Chapter 8 Introducing Algebraic Geometry

(Commutative Algebra 2) 8.1 Varieties 8.2 The Hilbert Basis Theorem 8.3 The Nullstellensatz: Preliminaries 8.4 The Nullstellensatz: Equivalent Versions and Proof 8.5 Localization 8.6 Primary Decomposition 8.7 Tensor Product of Modules Over a Commutative Ring 8.8 General Tensor Products

Chapter 9 Introducing Noncommutative Algebra

9.1 Semisimple Modules 9.2 Two Key Theorems 9.3 Simple and Semisimple Rings 9.4 Further Properties of Simple Rings, Matrix Rings, and Endomorphisms 9.5 The Structure of Semisimple Rings 9.6 Maschke’s Theorem 9.7 The Jacobson Radical 9.8 Theorems of Hopkins-Levitzki and Nakayama

Chapter 10 Introducing Homological Algebra

10.1 Categories 10.2 Products and Coproducts 10.3 Functors 10.4 Exact Functors 10.5 Projective Modules 10.6 Injective Modules 10.7 Embedding into an Injective Module 10.8 Flat Modules 10.9 Direct and Inverse Limits Appendix to Chapter 10

Supplement

S1 Chain Complexes S2 The Snake Lemma S3 The Long Exact Homology Sequence S4 Projective and Injective Resolutions S5 Derived Functors S6 Some Properties of Ext and Tor S7 Base Change in the Tensor Product

Solutions to Problems

Chapters 1–5 Chapters 6–10

Bibliography List of Symbols Index