# 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

## 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
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

### © 2000 by John A. Beachy and William D. Blair.

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

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