Book:Richard A. Dean/Elements of Abstract Algebra

From ProofWiki
Jump to: navigation, search

Richard A. Dean: Elements of Abstract Algebra

Published $1966$, Wiley International Edition

ISBN 0 471 20452 8.


Subject Matter


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