Book:Richard A. Dean/Elements of Abstract Algebra

Subject Matter

 * Abstract Algebra
 * Group Theory
 * Field Theory
 * Euclidean Domains
 * Polynomial Theory
 * Vector Spaces
 * Galois Theory

Contents

 * Preface


 * Chapter 0
 * 0.1 Arithmetic
 * 0.2 Sets
 * 0.3 Relations
 * 0.4 Functions and Mappings
 * 0.5 Operations and Operators
 * 0.6 Combinatorics


 * Prologue


 * Chapter 1: Groups
 * 1.1 Introduction
 * 1.2 Group Axioms
 * 1.3 Examples
 * 1.4 Basic Lemmas
 * 1.5 Isomorphism
 * 1.6 Permutation Groups
 * 1.7 Cyclic Groups
 * 1.8 Dihedral Groups
 * 1.9 Subgroups
 * 1.10 Homomorphisms
 * 1.11 Direct Products


 * Chapter 2: Rings
 * 2.1 Definitions
 * 2.2 Basic Lemmas
 * 2.3 Subrings
 * 2.4 Homomorphisms
 * 2.5 Integral Domains


 * Chapter 3: The Integers
 * 3.1 Introduction
 * 3.2 Order
 * 3.3 Order in Integral Domains
 * 3.4 Well-Ordered Sets
 * 3.5 The Integers
 * 3.6 Arithmetic in the Integers


 * Chapter 4: Fields
 * 4.1 Introduction
 * 4.2 Field of Quotients
 * 4.3 Subfields
 * 4.4 Homomorphism of Fields
 * 4.5 The Real Numbers
 * 4.6 The Complex Numbers


 * Chapter 5: Euclidean Domains
 * 5.1 Introduction
 * 5.2 The Euclidean Algorithm
 * 5.3 Arithmetic in Euclidean Domains
 * 5.4 Application to Groups


 * Chapter 6: Polynomials
 * 6.1 Introduction
 * 6.2 Polynomial Rings
 * 6.3 Polynomials Over a Field
 * 6.4 The Complex Numbers
 * 6.5 Special Properties of $F \left[{x}\right]$
 * 6.6 Factorization in $R \left[{x}\right]$
 * 6.7 Field of Quotients of $R \left[{x}\right]$
 * 6.8 Polynomials in Several Variables


 * Chapter 7: Vector Spaces
 * 7.1 Introduction
 * 7.2 Definition and Examples
 * 7.3 Subspaces
 * 7.4 Dependence and Basis
 * 7.5 Linear Transformations
 * 7.6 Solutions of Systems of Linear Equations
 * 7.7 Algebras


 * Chapter 8: Field Extensions and Finite Fields
 * 8.1 Construction of Field Extensions
 * 8.2 Classification of Extensions
 * 8.3 Transcendental Extensions
 * 8.4 Algebraic Extensions
 * 8.5 Finite Fields
 * 8.6 Simple Extensions
 * 8.7 Roots of Unity
 * 8.8 Wedderburn's Theorem


 * Chapter 9: Finite Groups
 * 9.1 Cauchy's Theorem
 * 9.2 $p$-Groups
 * 9.3 The Sylow Theorems
 * 9.4 Solvable Groups
 * 9.5 Abelian Groups


 * Chapter 10: Galois Theory
 * 10.1 Fundamental Theorem of Galois Theory
 * 10.2 Cyclotomic Fields and Cyclic Extensions
 * 10.3 Solution of Equations by Radicals
 * 10.4 Equations of 2nd and 3rd Degree
 * 10.5 The General Polynomial of $n$th Degree
 * 10.6 The Discriminant
 * 10.7 Symmetric Polynomials


 * Answers to Exercises
 * Symbols and Notations
 * Index