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

 

© 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


"Introduction" to Abstract Algebra | Abstract Algebra Online