# Contemporary Abstract Algebra

###### Joseph A. Gallian
Publisher:
Brooks/Cole Cengage
Publication Date:
2012
Number of Pages:
656
Format:
Hardcover
Edition:
8
Price:
185.95
ISBN:
978-1133599708
Category:
Textbook
BLL Rating:

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Richard Wilders
, on
02/1/2013
]

This old standard (the first edition came out 25 years ago) has been nicely refreshed. I have taught successfully (or so I hope) out of several editions of this text and have yet to find a better choice for a first undergraduate course in abstract algebra. It is written in a style which is both rigorous and welcoming. Anyone who knows Gallian will find his personality on almost every page.

The degree of difficulty and topic order of this text seems ideal for the majority of students. Careful choice of exercises and chapters on the part of the instructor can vary this difficulty level enough to make this text a safe choice for all but high-level honors sections. For example, Chapter 0 provides a nice introduction to the integers and equivalence relations. If your department offers a bridge course, you might skip this chapter.

Gallian treats several topics which most texts omit or treat in much less detail. Among these are Geometric Constructions, Finite Simple Groups, Frieze Groups and Crystallographic Groups, and Algebraic Coding Theory. While most of us won’t be able to cover all of these topics in to a two-semester course, they provide lots of options for an honors section or as starters for student honors projects.

According to its introduction, the new edition contains 200 new exercises, new examples and an updating of the quotations (which are wonderful), historical notes and biographies. Those features are among the many strengths of Gallian’s text and the updates are well done.

The answers to selected exercises are more than just answers providing what Gallian refers to as “skeleton solutions and hints.” Additionally the text now comes with most of the ancillaries we have come to expect from Calculus texts including:

Student Solutions Manual. Instructor’s Solution Manual. Online Solution Builder which allows instructors to create solutions printouts in PDF. Website with student resources including T/F questions with comments, flashcards, essays on learning abstract algebra.

The exercises range from simple computations to very interesting theoretical problems, many of which provide previews of theorems to come. The number of exercises on cyclic groups has gone from 69 to 86. Several of the added problems were simple computations of the sort I would assign when I used earlier editions. Here are two examples:

57) Determine the orders of the elements of D33 and how many there are of each.

72) Let a be a group element with |a| = 48. Find a divisor k of 48 such that

a) (langle a^{21} angle = langle a^{k} angle )

b) (langle a^{14} angle = langle a^{k} angle )

c) (langle a^{18} angle = langle a^{k} angle )

Here are a couple of more theoretical exercises that were added. From Chapter 14:

43) If (R) and (S) are principal ideal domains prove that (Roplus S) is also a principal ideal domain.

From Chapter 18:

44) Let F be a field and R be the integral domain in F[x] generated by x2 and x3 (that is, R is contained in every integral domain that contains x2 and x3).  Show  that F is not a unique factorization domain.

Several of the photos/images of mathematicians that appear at the end of most chapters have been updated. I quickly scanned back to the chapter on finite simple groups to see what had happened to the photos of Aschbacher, Gorenstein, and Thompson. Thompson’s photo had not changed, but Aschbacher’s is now much more current. Interestingly, Thompson’s appears to be a much older shot of him as a young man working with a student. It would be nice if the photos and quotes were dated, but that is a very small quibble.

Abstract Algebra 8th Edition is a very solid text which is suitable for a wide range of students. In addition, it now has all the bells and whistles we expect in a main-line calculus text. It is highly recommended. Indeed, even if you don’t choose to use it for your course, you should have a copy on hand to peruse for quotes, good examples and great mini-biographies.

Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences at North Central College in Naperville, IL.

PART I: INTEGERS AND EQUIVALENCE RELATIONS.
Preliminaries. Properties of Integers. Complex Numbers. Modular Arithmetic. Mathematical Induction. Equivalence Relations. Functions (Mappings). Exercises.
PART I: GROUPS.
1. Introduction to Groups.
Symmetries of a Square. The Dihedral Groups. Exercises. Biography of Neils Abel
2. Groups.
Definition and Examples of Groups. Elementary Properties of Groups. Historical Note. Exercises.
3. Finite Groups; Subgroups.
Terminology and Notation. Subgroup Tests. Examples of Subgroups. Exercises.
4. Cyclic Groups.
Properties of Cyclic Groups. Classification of Subgroups of Cyclic Groups. Exercises. Biography of J. J. Sylvester. Supplementary Exercises for Chapters 1-4.
5. Permutation Groups.
Definition and Notation. Cycle Notation. Properties of Permutations. A Check-Digit Scheme Based on D5. Exercises. Biography of Augustin Cauchy.
6. Isomorphisms.
Motivation. Definition and Examples. Cayley's Theorem. Properties of Isomorphisms.
Automorphisms. Exercises. Biography of Arthur Cayley.
7. Cosets and Lagrange's Theorem.
Properties of Cosets. Lagrange's Theorem and Consequences. An Application of Cosets to Permutation Groups. The Rotation Group of a Cube and a Soccer Ball. Exercises. Biography of Joseph Lagrange.
8. External Direct Products.
Definition and Examples. Properties of External Direct Products. The Group of Units Modulo n as an External Direct Product. Applications. Exercises.
Biography of Leonard Adleman. Supplementary Exercises for Chapters 5-8
9. Normal Subgroups and Factor Groups.
Normal Subgroups. Factor Groups. Applications of Factor Groups. Internal Direct Products. Exercises. Biography of Évariste Galois
10. Group Homomorphisms.
Definition and Examples. Properties of Homomorphisms. The First Isomorphism Theorem. Exercises. Biography of Camille Jordan.
11. Fundamental Theorem of Finite Abelian Groups.
The Fundamental Theorem. The Isomorphism Classes of Abelian Groups. Proof of the Fundamental Theorem. Exercises. Supplementary Exercises for Chapters 9-11.
PART III: RINGS.
12. Introduction to Rings.
Motivation and Definition. Examples of Rings. Properties of Rings. Subrings. Exercises. Biography of I. N. Herstein.
13. Integral Domains.
Definition and Examples. Fields. Characteristic of a Ring. Exercises. Biography of Nathan Jacobson.
14. Ideals and Factor Rings.
Ideals. Factor Rings. Prime Ideals and Maximal Ideals. Exercises.
Biography of Richard Dedekind. Biography of Emmy Noether. Supplementary Exercises for Chapters 12-14.
15. Ring Homomorphisms.
Definition and Examples. Properties of Ring Homomorphisms. The Field of Quotients.
Exercises.
16. Polynomial Rings.
Notation and Terminology. The Division Algorithm and Consequences. Exercises.
Biography of Saunders Mac Lane.
17. Factorization of Polynomials.
Reducibility Tests. Irreducibility Tests. Unique Factorization in Z[x]. Weird Dice: An Application of Unique Factorization. Exercises. Biography of Serge Lang.
18. Divisibility in Integral Domains.
Irreducibles, Primes. Historical Discussion of Fermat's Last Theorem. Unique Factorization Domains. Euclidean Domains. Exercises.
Biography of Sophie Germain. Biography of Andrew Wiles. Supplementary Exercises for Chapters 15-18.
PART IV: FIELDS.
19. Vector Spaces.
Definition and Examples. Subspaces. Linear Independence. Exercises. Biography of Emil Artin. Biography of Olga Taussky-Todd.
20. Extension Fields.
The Fundamental Theorem of Field Theory. Splitting Fields. Zeros of an Irreducible Polynomial. Exercises. Biography of Leopold Kronecker.
21. Algebraic Extensions.
Characterization of Extensions. Finite Extensions. Properties of Algebraic Extensions
Exercises. Biography of Irving Kaplansky.
22. Finite Fields.
Classification of Finite Fields. Structure of Finite Fields. Subfields of a Finite Field.
Exercises. Biography of L. E. Dickson.
23. Geometric Constructions.
Historical Discussion of Geometric Constructions. Constructible Numbers. Angle-Trisectors and Circle-Squarers. Exercises. Supplementary Exercises for Chapters 19-23.
PART V: SPECIAL TOPICS.
24. Sylow Theorems.
Conjugacy Classes. The Class Equation. The Probability That Two Elements Commute. The Sylow Theorems. Applications of Sylow Theorems. Exercises. Biography of Ludvig Sylow.
25. Finite Simple Groups.
Historical Background. Nonsimplicity Tests. The Simplicity of A5. The Fields Medal. The Cole Prize. Exercises. Biography of Michael Aschbacher. Biography of Daniel Gorenstein. Biography of John Thompson.
26. Generators and Relations.
Motivation. Definitions and Notation. Free Group. Generators and Relations. Classification of Groups of Order up to 15. Characterization of Dihedral Groups. Realizing the Dihedral Groups with Mirrors. Exercises. Biography of Marshall Hall, Jr..
27. Symmetry Groups.
Isometries. Classification of Finite Plane Symmetry Groups. Classification of Finite Group of Rotations in R³. Exercises.
28. Frieze Groups and Crystallographic Groups.
The Frieze Groups. The Crystallographic Groups. Identification of Plane Periodic Patterns. Exercises. Biography of M. C. Escher. Biography of George Pólya. Biography of John H. Conway.
29. Symmetry and Counting.
Motivation. Burnside's Theorem. Applications. Group Action. Exercises. Biography of William Burnside.
30. Cayley Digraphs of Groups.
Motivation. The Cayley Digraph of a Group. Hamiltonian Circuits and Paths. Some Applications. Exercises. Biography of William-Rowan Hamilton. Biography of Paul Erdös.
31. Introduction to Algebraic Coding Theory.
Motivation. Linear Codes. Parity-Check Matrix Decoding. Coset Decoding.
Historical Note: The Ubiquitous Reed-Solomon Codes. Exercises. Biography of Richard W. Hamming. Biography of Jessie MacWilliams. Biography of Vera Pless. 32. An Introduction to Galois Theory. Fundamental Theorem of Galois Theory. Solvability of Polynomials by. Radicals. Insolvability of a Quintic. Exercises. Biography of Philip Hall.
33. Cyclotomic Extensions.
Motivation. Cyclotomic Polynomials. The Constructible Regular n-gons. Exercises. Biography of Carl Friedrich Gauss. Biography of Manjul Bhargava.
Supplementary Exercises for Chapters 24-33.