**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**