1CHAPTER 1 Introduction to Abstract Algebra
235.2 Subspaces and Linear Independence
21.1 Historical Background
245.3 Basis and Dimension
31.2 Fundamental Concepts
255.4 Linear Transformations
41.3 Motivation and Scope
265.5 Eigenvalues and Eigenvectors
5CHAPTER 2 Sets, Relations, and Functions
27CHAPTER 6 Advanced Topics in Group Theory
62.1 Set Theory Fundamentals
286.1 Symmetry Groups
72.2 Equivalence Relations
296.2 Sylow Theorems
82.3 Functions and Mappings
306.3 Direct and Semidirect Products
9CHAPTER 3 Group Theory
316.4 Group Representations
103.1 Basic Definitions and Examples
32CHAPTER 7 Field Extensions and Galois Theory
113.2 Subgroups and Cosets
337.1 Field Extensions
123.3 Group Homomorphisms
347.2 Algebraic and Transcendental Extensions
133.4 Isomorphism Theorems
357.3 Splitting Fields and Algebraic Closure
143.5 Group Actions
367.4 Galois Theory: Basic Concepts
15CHAPTER 4 Rings and Fields
377.5 Solvability by Radicals
164.1 Definition and Examples of Rings
38Chapter 8 Applications of Abstract Algebra
174.2 Integral Domains and Fields
398.1 Cryptography and Coding Theory
184.3 Ring Homomorphisms
408.2 Algebraic Geometry
194.4 Ideals and Quotient Rings
418.3 Algebraic Number Theory
204.5 Polynomial Rings
428.4 Group Theory in Physics
21CHAPTER 5 Vector Spaces
43Glossary
225.1 Vector Space Definitions and Examples
44Index
