Abstract Algebra: Theory and Applications

Reviewed by Malik Barrett, Assistant Professor, Earlham College on 6/24/19

Comprehensiveness rating: 5 see less

Judson covers all of the basics one expects to see in an undergraduate algebra sequence. That is, some review from discrete math/intro to proofs (chapters 1-2), and elementary group theory including chapters on matrix groups, group structure, actions, and Sylow theorems.

The coverage of ring theory is slimmer, but still relatively "complete" for a semester of undergraduate study. Three chapters on rings, one on lattices, a chapter reviewing linear algebra, and three chapters on field theory with an eye towards three classical applications of Galois theory. I will note here that Judson avoids generators and relations.

The coverage is all fairly standard, with excepting the definition of Galois group (see accuracy), and the referencing system in the HTML version is extremely convenient. For example, Judson leverages HTML so that proofs are collapsed (but can be expanded) which allows him to clean up the presentation of each section and include full proofs of earlier results when useful as references. The index uses a similar approach, choosing to display a collapsed link to the first paragraph in which the term is used, which is often a formal definition. There are no pages displayed, but there is a google search bar to scan the book with. Given the searchability, the index style is an interesting choice.

Since Judson includes _a lot_ of Sage which he uses to expand, clarify, or apply theory from the text, a fairly standard presentation of the theory, and includes hints/solutions to selected exercises, the textbook is very comprehensive.

Content Accuracy rating: 4

I've noticed very few outright errors in the text proper. However, of primary note is Judson's non-standard (in my experience) definition of Galois group as the automorphism group Aut(E/F) of an arbitrary field extension E/F. He defines this before he's defined fixed fields (ala Artin), or normal/separable extensions. All of the exercises use this definition as well, and so I chose to (mostly) avoid the chapter on Galois theory in favor of a more standard presentation.

There _are_ some errors in the exercises, however, like the inclusion of unnecessary or irrelevant parts, or typos. But I came across very few of these in my problem sets.

Relevance/Longevity rating: 5

Modern applications are sprinkled throughout the text that informs the students of the value of the material beyond theoretical. Judson does this in practical ways given that Sage is such a big component of the book, and so there are many exercises and descriptions that stress this relevance.

Clarity rating: 4

Judson's writing is direct and effective. I find his style clean and easy to follow. However, there are instances where there are big jumps between what some beginning exercises assume and what was presented explicitly in the chapter which confused many of my students. For instance, there is a dearth of examples of how to compute minimal polynomials and extension degrees (and the subtleties involved), and so the instructor has to provide the strategies necessary to solve parts of the first two problems.

Consistency rating: 5

The book is consistent in language, tone, and style. The only inconsistencies I've noticed involve the occasional definition appearing inline (usually in a sentence motivating the definition) instead of set aside in a text box. Defined terms _are_ still shown in bold, though. Still, it can make it hard to locate the precise definition quickly by scanning the section, but happens so rarely I won't detract a point.

Modularity rating: 5

Judson is very direct, and so his chapters are very focused. Moreover, many sections are punctuated, perhaps including no more than several definitions and propositions along with a historical note. So it's quite easy to divide the material into tight, bite-sized portions along the sections of the book, with a few exceptions, i.e., sections that run -much- longer and denser than average, like the section on field automorphisms.

Many sections and some chapters are written in a way that relies minimally on previous material which allows one to omit them or change the order of presentation without too much fuss. For instance, it's easy to cover the material on matrix groups and symmetry (chapter 12) right after the intro coverage of groups (chapter 3) if you want more concrete examples. Or omit the chapters on integral domains (with some minimal adjustment), lattices, and linear algebra if one is making a push to fields and Galois theory.

Organization/Structure/Flow rating: 5

The text has a relatively linear progression, with some exceptions. The exceptions aren't detractions, though, and allow for modularity or digressions to applications.

Interface rating: 5

The UI of the text is amazingly clean and efficient. Google search makes scanning the book quick and easy, the collapsible table of contents and the sidebar makes jumping around in the text simple. Sage can be run on the page itself making the Sage section quite effective. One can even right-click on rendered LaTeX, like tables, and copy the underlying code (which is super convenient for Cayley tables).

Grammatical Errors rating: 5

I recall no major grammatical errors.

Cultural Relevance rating: 5

Judson sticks to the math, so the text is pretty impersonal. Even the historical notes are fact-based accounts.

Comments

I used the book for a year-long algebra sequence and was fairly happy with the outcome. Beyond the first two sections of the Galois theory chapter being too non-standard for my tastes, I had few complaints and will very likely use the text again. The problem bank is also very good and they generally complement the material from the chapters quite well.

Reviewed by Andrew Misseldine, Assistant Professor, Southern Utah University on 6/19/18

Comprehensiveness rating: 5 see less

This textbook is recommended for a upper division undergraduate course on abstract algebra and contains enough materials to cover a two-semester sequence, with particular emphasis placed on groups, rings, and fields. The group theory contains all the main topics of undergraduate algebra, including subgroups, cosets, normal subgroups, quotient groups, homomorphisms, and isomorphism theorems and introduces students to the important families of groups, with a particular emphasis on finite groups, such as cyclic, abelian, dihedral, permutation, and matrix groups. The textbook also includes more advanced topics such as structure of finite abelian groups, solvable groups, group actions, and Sylow Theory. The coverage of rings is equally comprehensive including the important topics of ideals, domains, fields, homomorphisms, polynomials, factorization, field extensions, and Galois Theory. The book is accompanied with a comprehensive index of topics and notation as well of solutions to selected exercises.

Content Accuracy rating: 5

The content of the textbook is very accurate, mathematically sound, and there are only a few errors throughout. The few errors which still exist can be reported to the author via email who appears to be very welcoming to suggestions or corrections from others. The author updates the textbook annually with corrections and additions.

Relevance/Longevity rating: 5

This textbook follows the classical approach to teaching groups, rings, and fields to undergraduate and will retain its value throughout the years as the theory and examples will not be changing. It is possible that some of the applications included, mostly related to computer science, could eventually become obsolete as new techniques are discovered, but this will probably not be too consequential to this text which is a math book and not a compute science textbook. The applications of algebra can still be interesting and motivating to the reader even if they are not the state-of-the-art. The author updates the textbook annually with corrections and is very welcoming to suggestions or corrections from others.

Clarity rating: 4

Overall, the textbook is very clear to read for those readers with the appropriate background of set theory, logic, and linear algebra. Proofs are particularly easy to follow and are well-written. The only real struggle here is in the homework exercises. Occasionally, the assumptions of the homework are not explicit which can lead to confusion for the student. This is often the fault that the exercises are collected for the entire chapter and not for individual sections. It can sometimes be a chore for instructors to assign regular homework because they might unintentionally assign an exercise which only involves vocabulary from an early section but whose proofs required theory from later in the chapter.

Consistency rating: 5

The author is consistent in his approach to both the theory and applications of abstract algebra, which matches in style many available textbooks on abstract algebra. In particular, the book's definitions and names of important theorems are in harmony with the greater body of algebraists. It is also consistent with its notation, although sometimes this notations deviates from the more popular notations and often fails to mention alternative notations used by others. A comprehensive notation index is included with references to the original introduction of the notation in the text. Regrettably, no similar glossary of terms exists except the index, which is should be sufficient for most readers.

Modularity rating: 4

The textbook is divided into chapters, sections, and subsections, with exercises and supplementary materials placed in the back of each chapter or at the end of the book. These headings and subheadings lead themselves naturally to how an instructor might parse the course material into regular lectures, but, dependent of the amount of detail desired by the instructor, these subsections do not often produce 50-minute lectures. The textbook's preface includes a dependency chart to help an instructor decide on the order of topics if time restricts complete coverage of the topics. The textbook could be easily adapted for a two semester sequence with the first semester covering groups and the second covering rings and fields or a single semester course which introduces both groups and rings while skipping the more advanced topics. The application chapters/sections can easily be included into the course or omitted from the course based upon the instructor's interest and background with virtually no interruption to the students. Some chapters include a section of "Additional Exercises" which include exercises about topic not covered in the textbook but adjacent to the topics introduced. Although these sections are prefaced by some explanation of the exploratory topic, rarely are these topics thorough explained which might leave student grossly confused and require the instructor to supplement the textbook on any exercises assigned from here.

Organization/Structure/Flow rating: 5

All sections follow the basic template of first introducing new definitions followed by examples, theorems, and proofs (although counterexamples are included, the presentation could benefit from additional counterexamples) and further definitions, examples, and theory are introduced as appropriate. Each chapter is concluded with a historical note, exercises for students, and references and suggested readings. Additionally, each chapter includes a section about programming in Sage relevant to the chapter contents with accompanying exercise, but this section is only available in the online version, not the downloadable or print versions. The first chapters review prerequisite materials including set theory and integers, which can be skipped by those students with a sufficient background without any loss. This book takes a "group-first" approach to introductory abstract algebra with rings, fields, vector spaces, and Boolean algebras introduced later. Throughout the textbook, in addition to the examples and theory, there are several practical applications of abstract algebra with a particular emphasis on computer science, such as cryptography and coding theory. These application sections/chapters can be easily included into the course without much extra preparation for the instructor or omitted at no real disruption to the student.

Interface rating: 5

This textbook was authored using PreTeXt, which designed for typesetting mathematical documents and allow them to be converted into multiple formats. This textbook is available in an online, downloadable pdf, and print version. All three versions have solid format, especially in regard to the mathematical typesetting and graphics. The online version is available in both English and Spanish, where the interface and readability are equally of high quality.

Grammatical Errors rating: 5

The textbook appears to be absent of regular grammatical or mathematical errors, although a few might be present. The few errors which still might exist can be reported to the author via email who appears to be very welcoming to suggestions or corrections from others. The author updates the textbook annually with corrections and additions. For the purposes of this review, the English version of the textbook was reviewed. The reviewed makes no claim about the quality of the grammar of the Spanish version which was translated by Antonio Behn from the author's original English version.

Cultural Relevance rating: 5

Culture is not really a concern for theoretical mathematics textbooks which focuses almost entire on mathematical content knowledge and theory and not so much on people or their relationships. The textbook is devoid of culturally insensitive of offensive materials. Many chapters end with historical notes about mathematicians who helped to develop the chapter's materials. These notes typically follow the traditional Western European narrative of abstract algebra's development and is fairly homogeneous. Efforts could be made to include a more diverse and international history of algebra beyond Europe. For example, there is no historical note about the Chinese Remainder Theorem other than a sentence to explain why its name includes the word "Chinese." The textbook, originally written in English, now includes a complete Spanish edition, which is a massive effort for any textbook to be more inclusive.

Comments

This has been one of my absolute favorite textbooks for teaching abstract algebra. In fact, I think Judson's book is a golden standard for what a high-quality, mathematical OER textbook should be. It has created using the very impressive PreTeXt. In addition to the different formats, this book includes SAGE exercises. It has enough material to fill the usual two-semester course in undergraduate abstract algebra.

Reviewed by Nicolae Anghel, Associate Professor, University of North Texas on 4/11/17

Comprehensiveness rating: 5 see less

This is a two-in-one book: a theoretical part and a computational part. Initially the OTL contained a 2014 version of the book, which only made tangential reference to the SAGE computational system. I downloaded from the author’s website the full, 2016 version, which eventually was also made into the OTL default.
The theoretical part of the book is certainly adequately comprehensive, covering evenly the proposed material, and being supported by judiciously chosen exercises. The computational part also seems to me comprehensive enough, however one should not take my word for it as this side exceeds my areas of expertise and interest.

Content Accuracy rating: 5

The parts that I checked, at random, were very accurate, so I have no reason to believe that the book was not entirely accurate. However, only after testing the book in the classroom, which I intend to do soon, can I certify this aspect.

Relevance/Longevity rating: 5

The material is highly relevant for any serious discussion on math curriculum, and will live as long as mankind does.

Clarity rating: 5

For me as instructor the book was very clear, however keep in mind that this was not the first source for learning the material. Things may be different for a beginning student, who sees the material for the first time. Again, a judgment on this should be postponed until testing the book in the classroom.

Consistency rating: 5

The book is consistent throughout, all the topics being covered thoroughly and meaningfully.

Modularity rating: 5

I have no substantive comments on this topic.

Organization/Structure/Flow rating: 5

The book, maybe a little too long for its own good, is divided into 23 chapters. The flow is natural, and builds on itself. The structure of each chapter is the same: After adequately presenting the material (conceptual definitions, theorems, examples), it proceeds to exercises, sometimes historical notes, references and further readings, to conclude with a substantial computational (based on SAGE syntax) discussion of the material, also including SAGE exercises. The applications to cryptography and coding theory highlight the practical importance of the material. I particularly liked the selection of exercises.

Interface rating: 5

Another big advantage of a free book is that the student does not have to print all of it, certainly not all of it at the same time. This is a big plus, since with commercial books most of the time a student buys a book and only a fraction of it is needed in a course.

Grammatical Errors rating: 5

Written in a conversational, informal style the book is by and large free of grammatical errors. There are about a dozen minor mistakes, such as concatenated words or repeated words.

Cultural Relevance rating: 5

The historical vignettes are sweet. Maybe adding pictures of the mathematicians involved would not be a bad thing.

Comments

I liked the book, but I like more the concept of free access to theoretical and practical knowledge. Best things in life should essentially be free: air, water, …, education. I will make an effort to use open textbooks whenever possible.

Reviewed by Daniel Hernández, Assistant Professor, University of Kansas on 8/21/16

Comprehensiveness rating: 5 see less

This book is introductory, and covers the basic of groups, rings, fields, and vector spaces. In addition, it also includes material on some interesting applications (e.g., public key cryptography). In terms of covering a lot of topics, the book is certainly comprehensive, and contains enough material for at least a year-long course for undergraduate math majors. A "dependency chart" in the preface should be very useful when deciding on what path to take through the text.

One noteworthy feature of this book is that it incorporates the open-source algebra program Sage. While the .pdf copy I found through the OTN website only included a not-very-serious discussion of Sage at the end of most exercise sets, the online textbook found at

http://abstract.pugetsound.edu/aata/

appears to contain a much more substantial discussion of how to use Sage to explore the ideas in this book. I admit that I didn't explore this feature very much.

Content Accuracy rating: 5

Though I have not checked every detail (the book is quite long!), there do not appear to be any major errors.

Relevance/Longevity rating: 5

The topics covered here are basic, and will therefore not require any real updates.

The book is also written in such a way that it should be easy to include new sections of applications.

Clarity rating: 5

I would say that this this book is well-written. The style is somewhat informal, and there are plenty of illustrative examples throughout the text. The first chapter also contains a brief discussion of what it means to write and read a mathematical proof, and gives many useful suggestions for beginners.

Through I didn't read every proof, in the ones I did look at, the arguments convey the key ideas without saying too much. The author also maintains the good habit of explicitly recalling what has been proved, and pointing out what remains to be done. In my experience, it is this sort of mid-proof "recap" is helpful for beginners.

Consistency rating: 5

The terminology in this text is standard, and appears to be consistent.

Modularity rating: 5

Each chapter is broken up into subsections, which makes it easy to for students to read, and for instructors to assign reading. In addition, this book covers modular arithmetic, which makes it even more "modular" in my opinion!

Organization/Structure/Flow rating: 4

It seems like there is no standard way to present this material. While the author's choices are perfectly fine, my personal bias would have been to discuss polynomial rings and fields earlier in the text.

Interface rating: 5

The link on page v to

abstract.pugetsound.edu

appears to be broken.

My browser also had some issues when browsing the Sage-related material on the online version of this text, but this may be a personal problem.

Grammatical Errors rating: 5

I did not notice any major grammatical errors.

Cultural Relevance rating: 5

I'm not certain that this question is appropriate for a math textbook. On the other hand, I'll take this as an opportunity to note that the historical notes that appear throughout are a nice touch.

Comments

The problem sets appear to be substantial and appropriate for a strong undergraduate student. Also, many sections contain problems that are meant to be solved by writing a computer program, which might be of interest for students studying computer science.

I am also slightly concerned that the book is so long that students may find it overwhelming and hard to sift through.