Abstract Algebra: Structures and Applications (Hardcover)

SET THEORY
Sets and Functions
The Cartesian Product; Operations; Relations
Equivalence Relations
Partial Orders

NUMBER THEORY
Basic Properties of Integers
Modular Arithmetic
Mathematical Induction

GROUPS
Symmetries of the Regular n-gon
Introduction to Groups
Properties of Group Elements
Symmetric Groups
Subgroups
Lattice of Subgroups
Group Homomorphisms
Group Presentations
Groups in Geometry
Diffie-Hellman Public Key
Semigroups and Monoids

QUOTIENT GROUPS
Cosets and Lagrange’s Theorem
Conjugacy and Normal Subgroups
Quotient Groups
Isomorphism Theorems
Fundamental Theorem of Finitely Generated Abelian Groups

RINGS
Introduction to Rings
Rings Generated by Elements
Matrix Rings
Ring Homomorphisms
Ideals
Quotient Rings
Maximal and Prime Ideals

DIVISIBILITY IN COMMUTATIVE RINGS
Divisibility in Commutative Rings
Rings of Fractions
Euclidean Domains
Unique Factorization Domains
Factorization of Polynomials
RSA Cryptography
Algebraic Integers

FIELD EXTENSIONS
Introduction to Field Extensions
Algebraic Extensions
Solving Cubic and Quartic Equations
Constructible Numbers
Cyclotomic Extensions
Splitting Fields and Algebraic Closures
Finite Fields

GROUP ACTIONS
Introduction to Group Actions
Orbits and Stabilizers
Transitive Group Actions
Groups Acting on Themselves
Sylow’s Theorem
A Brief Introduction to Representations of Groups

CLASSIFICATION OF GROUPS
Composition Series and Solvable Groups
Finite Simple Groups
Semidirect Product. Classification Theorems
Nilpotent Groups

MODULES AND ALGEBRAS
Boolean Algebras
Vector Spaces
Introduction to Modules
Homomorphisms and Quotient Modules
Free Modules and Module Decomposition
Finitely Generated Modules over PIDs, I
Finitely Generated Modules over PIDs, II
Applications to Linear Transformations
Jordan Canonical Form
Applications of the Jordan Canonical Form
A Brief Introduction to Path Algebras

GALOIS THEORY
Automorphisms of Field Extensions
Fundamental Theorem of Galois Theory
First Applications of Galois Theory
Galois Groups of Cyclotomic Extensions
Symmetries among Roots; The Discriminant
Computing Galois Groups of Polynomials
Fields of Finite Characteristic
Solvability by Radicals

MULTIVARIABLE POLYNOMIAL RINGS
Introduction to Noetherian Rings
Multivariable Polynomial Rings and Affine Space
The Nullstellensatz
Polynomial Division; Monomial Orders
Grobner Bases
Buchberger’s Algorithm
Applications of Grobner Bases
A Brief Introduction to Algebraic Geometry

CATEGORIES
Introduction to Categories
Functors

Projects appear at the end of each chapter.