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A Book of Abstract Algebra

Charles C. Pinter
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

The unusual and attractive feature of this book is that over half of the space is given to problem sequences. The book is organized as a collection of 33 short chapters, each one having a narrative section that deals with a single problem or concept, and a lengthy exercise section that further develops the ideas of the chapter. The present book is an unaltered reprint of the 1990 second edition.

The choice of topics and their treatment is very conventional, but the author has thought carefully about what the most important ideas are, and has highlighted those in the narratives. The chapters on Galois theory are especially good and really bring out the importance of field automorphisms, which I had never really appreciated before (having always gotten bogged down in the solvable-groups part of the theory).

Very Good Feature: topics are not introduced until they are needed, so you don’t have to wonder (for more than a few pages) why the author is telling you all this. For example, vector spaces are treated in the middle of field theory, because they are needed for field extensions.

On the down side, the exercises are not challenging; if a difficult problem needs to be solved, it is broken down into a series of simple steps. Most of the exercises are proofs rather than drill, and if you work through them all you will have a very thorough knowledge of the subject, but the process will not stretch you very much.

An even more stark book is Clark’s Elements of Abstract Algebra. This also strips the subject down to its most essential parts, but has only a few exercises. A more expansive survey book where everything is worked out, and that also has a good bit broader coverage, is Birkhoff & Mac Lane’s classic A Survey of Modern Algebra.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

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
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
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
Invertibles and Zero-Divisors
Integral Domain

18 Ideals and Homomorphisms

19 Quotient Rings
Construction of Quotient Rings
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
Characterization of Ζ Up to Isomorphism
Mathematical Induction
Division Algorithm

22 Factoring into Primes
Ideals of Ζ
Properties of the GCD
Relatively Prime Integers
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
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
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