A Computational Introduction to Number Theory and Algebra

Reviewed by William McGovern, Professor, University of Washingon on 8/21/16

Comprehensiveness rating: 5 see less

As promised by the title, the book gives a very nice overview of a side range of topics in number theory and algebra (primarily the former, but with quite a bit of attention to the latter as well), with special emphasis to the areas in which computational techniques have proved useful. There is a very good index and glossary and a good review of notation and basic facts in the first chapter.

Content Accuracy rating: 5

The content is very accurate ad up to date. I see no signs of bias.

Relevance/Longevity rating: 5

The format of the book makes it especially easy to update as advances in the subjects occur, particularly computational advances. References are given to websites as well as books.

Clarity rating: 5

The prose is very lucid and easy to follow. Many examples are given and difficult ideas are introduced gradually. The many relationships between number theory and algebra are explored in detail, each subject yielding important insights into and applications of the other. No jargon is used and terminology is carefully explained.

Consistency rating: 5

The book has a very consistent framework and a nice flow from one chapter to the next. As mentioned above, relationships between the two subjects of the title are emphasized.

Modularity rating: 5

The book is nicely broken up into manageable sections that would fit well into a lecture course. Interdependences among chapters are clearly indicated.

Organization/Structure/Flow rating: 5

The topics are presented clearly and logically with relationships among them clearly pointed out and discussed in detail.

Interface rating: 5

All pages display very well on my screen, with no legibility or distortion issues that I could see.

Grammatical Errors rating: 5

The grammar seems fine.

Cultural Relevance rating: 5

This is not relevant for a mathematics text, but I saw nothing that would be offensive to a reader of any ethnic background.

Comments

I would be happy to teach a course out of this book.

Reviewed by Michelle Manes, Associate Professor, University of Hawaii on 8/21/16

Comprehensiveness rating: 4 see less

The text is so comprehensive that it feels overwhelming. The author wanted to include all of the mathematics required beyond a standard calculus sequence. However, the mathematical maturity required to read and learn from this text is quite high.

The first two chapters cover much of a standard undergraduate course in number theory, built up from scratch. However, it almost completely lacks numerical examples and computational practice for the students, which would give those new to the material time and experience in which to digest, assimilate, and understand the material. I would think that a book targeted at this level of mathematical sophistication would assume students are comfortable with (for example) the most basic notions of group theory or the idea of equivalence classes.

I can't imagine an appropriate audience for this text: one with the ability to read and work entirely at this abstract level but without any (or most) of the mathematical preparation provided in at least half the chapters.

Content Accuracy rating: 5

I found no mathematical errors. The mathematical presentation is rigorous, clear, and well-explained. It can be terse at times, skipping steps and making conceptual leaps that will be challenging for all but the very best students.

Relevance/Longevity rating: 4

The book covers both standard background that will always be relevant for these topics: the number theory and algebra background, the probability theory. The computational chapters use pseudocode, so they will not be quickly outdated when new languages become fashionable. Most of the algorithms studied are quite "classical" (as much as that makes sense for computer science), with modern ideas and developments usually relegated to "Notes" at the end of the computational chapters. This will, of course, become outdated with new research in computer science. But any faculty member who keeps up with the relevant research will be able to mention new developments to students, and it will not interrupt the flow of the ideas at all.

Clarity rating: 4

The book is exceedingly well written, though it is at a very high level. It is not "friendly" or "chatty" as you will find with many number theory books targeted to undergraduates. For many students this will detract from clarity because they do not yet have the mathematical sophistication to work at this level.

Consistency rating: 5

The book does an excellent job of consistency of notation. For example, it starts with a development of number theory concepts, and develops notation for residue classes in the integers modulo n. Later in the chapters on groups and rings, this same notation is used in more general situations. Whenever there is the potential for confusion (for example, in using "a mod b" as a binary operation as is common in computer science versus using "a is congruent to x mod b" as is more standard in mathematics) the author is careful to point out the dual meanings and to warn the reader that there is some overloading of terminology. It is unavoidable that this will happen in any book that treats both subjects seriously, and the author is careful with notation and keeps potential confusion to a minimum.

Modularity rating: 4

The book has 21 chapters, each with several sections. Most, but not all, sections end with a set of exercises. Essential exercises are underlined (a very nice feature!) and optional sections are indicated with an asterisk. What would be helpful would be some suggested paths through the text for various purposes. I don't think it would be appropriate in any class to start at Chapter 1 and and work through all (or even most) of the content. I imagine that most classes would skip the background material and head straight for the computational chapters, with the background there "as needed" for the students.

Organization/Structure/Flow rating: 4

My main comment about the structure is that the mathematics chapters and the computational chapters seem to be separated. For example, the chapter on "Congruences" covers a tremendous amount of number theory, not all of which falls naturally (in my mind) under that heading. Chapter 1 has a section on "Ideals and greatest common divisors," but Euclid's Algorithm is not tackled until Chapter 4 (a more computational chapter). As I read, I often felt "now we are doing mathematics... now we are concerned with computational questions." There are natural places of overlap (like Euclid's algorithm), and they are separated rather than treated more holistically.

Interface rating: 4

I read a standard PDF file. There were a few hyperlinks (from the table of contents to section headings, for example), but not much else in the way of interface. Everything was rendered clearly.

Grammatical Errors rating: 5

I found no errors.

Cultural Relevance rating: 3

There is a lot of interesting history and "cultural" notes in the computing chapters, and almost none in the more mathematical chapters. A student who studied from this text would miss a lot of the standard "mathematics culture" communicated in a more traditional number theory course.

Comments

My primary comment is that I cannot pin down the audience for this book. I could not use this in an undergraduate number theory class; it is at far too high a level and moves far too quickly. I could not use it in a graduate number theory class; it assumes no background at all and does not do some standard topics. I suppose it would be useful for self-study by a very advanced student who already knew a good deal of mathematics and wanted to explore the computational side. I do think that the title "A Computational Introduction to Number Theory and Algebra" is misleading at best. Lacking numerical examples (for examples, students never actually do any "clock arithmetic" type calculations when introduced to the integers mod n) and with a focus only on abelian groups and commutative rings with unity, the book is simultaneously too sophisticated and not sophisticated enough for my use. It also has a bit of a "joyless" feel in the mathematics, with a development of ideas and a writing style that never brings students into any part of the discovery of the mathematics.

Reviewed by Emily Witt, Assistant Professor, University of Kansas on 8/21/16

Comprehensiveness rating: 5 see less

This text is an introduction to number theory and abstract algebra; based on its presentation, it appears appropriate for students coming from computer science. The book starts with basic properties of integers (e.g., divisibility, unique factorization), and touches on topics in elementary number theory (e.g., arithmetic modulo n, the distribution of primes, discrete logarithms, primality testing, quadratic reciprocity) and abstract algebra (e.g., groups, rings, ideals, modules, fields and vector spaces, some linear algebra, polynomial rings and their quotients). The book also includes an introduction to probability. This, and other topics, are tools for interesting computational applications. The Table of Contents indicates a few sections that are not required for future material. The text includes an effective index.

Content Accuracy rating: 5

From my research in writing this review, I have not come across any major errors. The author has a list of errata on his webpage.

Relevance/Longevity rating: 5

The book appears to be up-to-date, and includes some interesting applications of theoretical material to topics relevant in cryptography (e.g., the RSA cryptosystem, and primality testing).

Clarity rating: 4

The presentation of topics is accurate, and starts "from scratch." All material necessary in future sections is included in the appropriate section. Some sections are terse, and an instructor may want to supplement the theoretical exercises with some more computational ones. The Euclidean Algorithm is presented after sections on solving linear congruences modulo n, and the Chinese Remainder Theorem; applications of the Euclidean Algorithm to these topics are presented later. An instructor may want students to become comfortable with these topics initially through computations, using the Euclidean Algorithm. We should also point out that mathematical induction is a prerequisite for this text, and some of the material is presented using pseudocode, which is different than many texts on these topics.

Consistency rating: 5

The book's terminology and mathematical frameworks appear to be consistent.

Modularity rating: 3

Since the book is quite long, an instructor for a one-semester course would need to choose specific topics from the text to cover. There are a few sections indicated that are not required for future material. However, even among the remaining sections, an instructor would need to carefully choose sections that include all necessary prerequisite material. Depending on a course's focus, this could be done fairly easily.

Organization/Structure/Flow rating: 3

Each chapter is written in a logical manner, referencing previous material as needed. The book jumps from chapters on purely algebraic topics to those focused on applications. For this reason, an instructor may want to choose certain sections in a chapter to cover as prerequisites for an application, instead of covering the material linearly.

Interface rating: 5

There do not appear to be any problems with the interface of the PDF of the text.

Grammatical Errors rating: 5

There do not appear to be major grammatical errors in the text.

Cultural Relevance rating: 5

Due to the topics in this text, this question does not appear to be applicable.