ABSTRACT ALGEBRA: OnLine Study Guide, Table of Contents
ABSTRACT ALGEBRA
ONLINE STUDY GUIDE
These online notes are for students
who are working through our textbook
Abstract Algebra.
The notes are focused on solved problems,
which are numbered consecutively,
beginning with the exercises in the text.
Our intention is to help students learn how to do proofs
as well as computations.
There are also some "lab" questions on groups, based on a Java applet
Groups15
written by
John Wavrik
of UCSD.
INTRODUCTION
 Some history
 Biographies
(from the MacTutor History of Mathematics archives)
 Some help with logic and proofs
 Links to some algebra sites
Chapter 1: INTEGERS

Introduction
 1.1
Divisors
Problems
Solutions
 1.2
Primes
Problems
Solutions
 1.3
Congruences
Problems
Solutions
 1.4
Integers Modulo n
Problems
Solutions

Review problems
Solutions
Chapter 2: FUNCTIONS

Introduction
 2.1
Functions
Problems
Solutions
 2.2
Equivalence relations
Problems
Solutions
 2.3
Permutations
Problems
Solutions

Review problems
Solutions
Chapter 3: GROUPS

Introduction
 3.1
Definition of a Group
Problems
Solutions
 3.2
Subgroups
Problems
Solutions
 3.3
Constructing Examples
Problems
Solutions

Some group multiplication tables
 3.4
Isomorphisms
Problems
Solutions
 3.5
Cyclic Groups
Problems
Solutions
 3.6
Permutation Groups
Problems
Solutions
 3.7
Homomorphisms
Problems
Solutions
 3.8
Cosets, Normal Subgroups, and Factor Groups
Problems
Solutions

Review problems
Solutions
Chapter 4: POLYNOMIALS

Introduction
 4.1
Fields; Roots of Polynomials
 4.2
Factors
 4.3
Polynomials with Integer Coefficients
 4.4
Existence of Roots

Review problems
Solutions
Chapter 5: COMMUTATIVE RINGS
 5.1
Commutative rings; Integral Domains
 5.2
Ring Homomorphisms
 5.3
Ideals and Factor Rings
 5.4
Quotient Fields

Review problems
Solutions
Chapter 6: FIELDS
 6.1
Algebraic Elements
 6.2
Finite and Algebraic Extensions
 6.3
Geometric Constructions
 6.4
Splitting fields
Problems
Solutions
 6.5
Finite Fields
Problems
Solutions
 6.6
Irreducible Polynomials over Finite Fields
 6.7
Quadratic Reciprocity

Review problems
Solutions
Chapter 7: STRUCTURE OF GROUPS

Introduction
 7.1
Isomorphism Theorems; Automorphisms
Problems
Solutions
 7.2
Conjugacy
Problems
Solutions
 7.3
Groups Acting on Sets
Problems
Solutions
 7.4
The Sylow Theorems
Problems
Solutions
 7.5
Finite Abelian Groups
Problems
Solutions
 7.6
Solvable Groups
Problems
Solutions
 7.7
Simple Groups
Problems
Solutions
Chapter 8: GALOIS THEORY

Introduction
 8.1
The Galois Group of a Polynomial
Problems
Solutions
 8.2
Multiplicity of Roots
Problems
Solutions
 8.3
The Fundamental Theorem of Galois Theory
Problems
Solutions
 8.4
Solvability by Radicals
Problems
Solutions
 8.5
Cyclotomic Polynomials
 8.6
Computing Galois Groups
Chapter 9: UNIQUE FACTORIZATION
 9.1
Principal Ideal Domains
 9.2
Unique Factorization Domains
 9.3
Some Diophantine Equations
INDEX
 Index of Definitions
 Index of Theorems
If you would like a printed copy of the solved problems,
rather than printing the html pages
you should download and print one of these supplements,
which are typeset using LaTex and are available
in either pdf or postscript format.
 Abstract Algebra: A Study Guide for Beginners
 Abstract Algebra: Review Problems on Groups and Galois Theory
"Introduction" to Abstract Algebra

Abstract Algebra Online