1 Why Abstract Algebra?
History of Algebra
New Algebras
Algebraic Structures
Axioms and Axiomatic Algebra
Abstraction in Algebra
2 Operations
Operations on a Set
Properties of Operations
3 The Definition of Groups
Groups
Examples of Infinite and Finite Groups
Examples of Abelian and Nonabelian Groups
Group Tables
Theory of Coding: Maximum-Likelihood Decoding
4 Elementary Properties of Groups
Uniqueness of Identity and Inverses
Properties of Inverses
Direct Product of Groups
5 Subgroups
Definition of Subgroup
Generators and Defining Relations
Cayley Diagrams
Center of a Group
Group Codes; Hamming Code
6 Functions
Injective, Subjective, Bijective Function
Composite and Inverse of Functions
Finite-State Machines
Automata and Their Semigroups
7 Groups of Permutations
Symmetric Groups
Dihedral Groups
An Application of Groups to Anthropology
8 Permutations of a Finite Set
Decomposition of Permutations into Cycles
Transpositions
Even and Odd Permutations
Alternating Groups
9 Isomorphism
The Concept of Isomorphism in Mathematics
Isomorphic and Nonisomorphic Groups
Cayley’s Theorem
Group Automorphisms
10 Order of Group Elements
Powers/Multiples of Group Elements
Laws of Exponents
Properties of the Order of Group Elements
11 Cyclic Groups
Finite and Infinite Cyclic Groups
Isomorphism of Cyclic Groups
Subgroups of Cyclic Groups
12 Partitions and Equivalence Relations
13 Counting Cosets
Lagrange’s Theorem and Elementary Consequences
Survey of Groups of Order ≤10
Number of Conjugate Elements
Group Acting on a Set
14 Homomorphisms
Elementary Properties of Homomorphisms
Normal Subgroups
Kernel and Range
Inner Direct Products
Conjugate Subgroups
15 Quotient Groups
Quotient Group Construction
Examples and Applications
The Class Equation
Induction on the Order of a Group
16 The Fundamental Homomorphism Theorem
Fundamental Homomorphism Theorem and Some Consequences
The Isomorphism Theorems
The Correspondence Theorem
Cauchy’s Theorem
Sylow Subgroups
Sylow’s Theorem
Decomposition Theorem for Finite Abelian Groups
17 Rings: Definitions and Elementary Properties
Commutative Rings
Unity
Invertibles and Zero-Divisors
Integral Domain
Field
18 Ideals and Homomorphisms
19 Quotient Rings
Construction of Quotient Rings
Examples
Fundamental Homomorphism Theorem and Some Consequences
Properties of Prime and Maximal Ideas
20 Integral Domains
Characteristic of an Integral Domain
Properties of the Characteristic
Finite Fields
Construction of the Field of Quotients
21 The Integers
Ordered Integral Domains
Well-ordering
Characterization of Ζ Up to Isomorphism
Mathematical Induction
Division Algorithm
22 Factoring into Primes
Ideals of Ζ
Properties of the GCD
Relatively Prime Integers
Primes
Euclid’s Lemma
Unique Factorization
23 Elements of Number Theory (Optional)
Properties of Congruence
Theorems of Fermat and Euler
Solutions of Linear Congruences
Chinese Remainder Theorem
Wilson’s Theorem and Consequences
Quadratic Residues
The Legendre Symbol
Primitive Roots
24 Rings of Polynomials
Motivation and Definitions
Domain of Polynomials over a Field
Division Algorithm
Polynomials in Several Variables
Fields of Polynomial Quotients
25 Factoring Polynomials
Ideals of F[x]
Properties of the GCD
Irreducible Polynomials
Unique factorization
Euclidean Algorithm
26 Substitution in Polynomials
Roots and Factors
Polynomial Functions
Polynomials over Q
Eisenstein’s Irreducibility Criterion
Polynomials over the Reals
Polynomial Interpolation
27 Extensions of Fields
Algebraic and Transcendental Elements
The Minimum Polynomial
Basic Theorem on Field Extensions
28 Vector Spaces
Elementary Properties of Vector Spaces
Linear Independence
Basis
Dimension
Linear Transformations
29 Degrees of Field Extensions
Simple and Iterated Extensions
Degree of an Iterated Extension
Fields of Algebraic Elements
Algebraic Numbers
Algebraic Closure
30 Ruler and Compass
Constructible Points and Numbers
Impossible Constructions
Constructible Angles and Polygons
31 Galois Theory: Preamble
Multiple Roots
Root Field
Extension of a Field
Isomorphism
Roots of Unity
Separable Polynomials
Normal Extensions
32 Galois Theory: The Heart of the Matter
Field Automorphisms
The Galois Group
The Galois Correspondence
Fundamental Theorem of Galois Theory
Computing Galois Groups
33 Solving Equations by Radicals
Radical Extensions
Abelian Extensions
Solvable Groups
Insolvability of the Quintic
Appendix A: Review of Set Theory
Appendix B: Review of the Integers
Appendix C: Review of Mathematical Induction
Answers to Selected Exercises
Index