Virtually everyone who has taken or
taught an undergraduate abstract algebra course knows the order of topics
is groups, rings, then fields. But have you ever thought about why we do
the topics in this order? Is it because the list of axioms for groups is
the shortest, and the list of field axioms is the longest? Surely
something with fewer axioms must be easier to understand, right? This book
challenges this conventional thinking. The motivation for doing so is the
premise that rings are inherently easier to understand than groups, and
that examples of rings familiar to students are quite plentiful. As such
this book begins with an extensive study of rings, then discusses groups,
and finally fields. While I was quite skeptical of this approach at first
(having been taught these topics in the "standard" order), I was
quickly won over by the book for this and many other reasons.The book is very complete, containing more than enough material for a two semester course in undergraduate abstract algebra (and weighing in at over 600 pages). Even though there was a great deal of material presented, I found the book to be very well organized. It is divided into large Sections by major topic, such as "Ring Homomorphisms and Ideals," "Groups," and "Vector Spaces and Field Extensions." Each Section is divided into Chapters by main concepts (typically 15 to 20 pages), which are then further subdivided by specific topic. This system made it easy to locate any subject in the table of contents.
The book begins (as most such books do) with a Chapter on preliminaries covering basic properties of the natural numbers, mathematical induction, well ordering, and the axiomatic method. I like the exercises at the end of this Chapter, particularly those on induction, because they provide the students with some interesting contexts (the triangle inequality, complete graphs, the binomial theorem) in which to apply the ideas discussed. The remaining Chapters in this Section discuss in depth the properties of the integers, then the integers modulo n, and finally polynomials with rational coefficients (presented in order of increasing abstractness), without mention of the word ring or the ring axioms. Once the students feel comfortable with these examples, the general notion of a ring is introduced in the next Section and the book takes off from here.
I agree with the authors' premise that rings are a better place to start in a first abstract algebra course than groups. Most of the examples of groups that we give students are also rings, and it can be confusing to the students to remember which operation they using to form a group. Additionally, I believe students benefit from a little more mathematical maturity before trying to understand standard examples of non-abelian groups (dihedral and symmetric groups, for instance). Contrast these with standard examples of rings: the integers, the integers modulo n, spaces of polynomials over a field, spaces of functions over a field, and matrices over a field. These are all ideas with which most math majors will have worked a great deal; calling them "rings" just puts familiar objects in a different light.
One of my only negative comments about the book is that it covers a great deal of ring theory (homomorphisms, ideals, integral domains, factorization and UFDs, PIDs, Euclidean domains, maximal and prime ideals, the Chinese Remainder Theorem) before even mentioning the word "group." An instructor following this text, it seems, would need to skip around in order to fit groups into the first semester of an abstract algebra sequence. While I agree that groups are more difficult to understand for most students, I still think they are an important concept which belongs in a first semester course.
There are a lot of things that I like about this book. At the end of each Chapter there is a Chapter Summary, and a Section Summary at the end of each Section. I really think these are well written and will help students to see the big picture. The book definitely seems to be written for students instead of instructors. It gives the motivation behind each discussion and goes to great lengths to explain each idea and theorem in detail — the chapters on constructibility seemed particularly well done to me. The exercises at the end of each Chapter are well written and thoughtful. There are also little exercises sprinkled throughout the text to aid in student understanding of statements and proofs (things which a mathematician would know to work out for himself, but an undergraduate student would not). All in all it seems that a lot of thought went into this book, resulting in a comprehensive, well-written, readable book for undergraduates first learning abstract algebra. [Frederick Butler]
Kai Lai Chung is an eminent probabilist, now
emeritus professor at Stanford. Born in Hangzhou, China, in 1917, he
graduated in 1940 from Tsinghua University at Kunming, where he studied
with Pao Lu Hsu (Xu Bao-lu). He came to the United States in 1945, and
obtained his Ph.D. at Princeton in 1947, under Bochner.
Chung has published many books, including undergraduate- and graduate-level textbooks in probability. I have had the pleasure of teaching an undergraduate course in probability from his book Elementary Probability Theory with Stochastic Processes, now in a fourth edition with new co-author Farid AitSahlia (Springer, 2003).
Chance and Choice: Memorabilia is a collection of Chung's papers, some of them research papers, some expository, some reviews. Most are connected in one way or another with the theory of Markov chains, though there are excursions into combinatorics. Even Jacobian theta-functions make an appearance. Most of the papers are in English, three are in French and one is in Italian. The contents and tragic circumstances of the one paper in Chinese are explained in the preceding paper in English. (The reader should be warned that the printer has interchanged page 2 and page 3. I can't resist pointing out here that the fundamental summation (8) that Chung uses in this first paper [p. 2] is an instance of the formula
which ought to be as familiar as the corresponding integration formula, but somehow is not. Here kn is the "falling power":
As Prof. Chung frequently remarks, "Notation can be incredibly important!")
There are occasional minor infelicities of language, hard to avoid entirely for a writer whose native tongue is not English. (For example, on p. 269 he refers to a "splightly author" — presumably intending "sprightly".) Despite these, Chung's writing is literate, elegant, wise, humane. He takes the reader into his confidence, explaining ideas, motivation, and circumstances. There are frequent aperçus. For example (p. 141), the article "Markov chain must have a beginning" begins, "The proposition quoted in the title above is a folklore. It can be argued as follows. If a Markov chain has run an infinitely long time, then every 'state' will have become a 'steady state', and nothing will happen any more. This is physicists' talk, which sounds apt for the occasion."
There are old photos of Chung with Cramér, Feller, Erdös, and others.
You probably won't want to buy this book unless you are a specialist in probability, or intend to become one. Nevertheless, it is well worth browsing through in the library. Look at the last article, "Mathematics and Applications". Has Chung overstated the case for the non-usefulness of mathematics? See what you think. [Stacy Langton]
Note: Prof. Chung has kindly informed me that his birthplace was Shanghai, not Hangzhou; and that his advisor at Princeton was Cramér, not Bochner. The meaning of the caption to photo number 13 of the book under review, which describes Hangzhou as "the author's native town," is that it is the town of his ancestors. I regret the errors. [Stacy Langton]
Somebody once said of opera, "You either love it
or you don't understand it." The same could be said of the calculus of
variations. Many, if not most mathematicians have never studied the
subject, and so could hardly be expected to love it. Bruce van Brunt shows
his love of the subject in his new book The Calculus of Variations,
part of the Universitext series from Springer.
All accounts of the calculus of variations start from the same foundation, the Euler-Lagrange equation, a differential equation that provides a condition necessary for a curve to be an optimal curve. Van Brunt takes the practical approach and makes the most of this necessary condition, rather than following a more theoretical tack and dwelling on sufficient conditions. In fact, this one emphasis can be used as a kind of litmus test to separate those books aimed at engineers and undergraduates from those written for mathematicians and graduate students. This book is crafted for engineers and undergraduates.
Van Brunt gives us a nice historical introduction to the calculus of variations. The topic has a long, if discontinuous history going to the Bernoulli brothers in the 1690s. He quotes a 1927 textbook that the subject "attracted a rather fickle attention at more or less isolated intervals in its growth." The giants of the 17th and 18th centuries, Newton, Leibniz, Euler, Lagrange, Legendre, gave us the first principles and the classical examples. It grew more slowly in the 19th century, only to be reawakened by Hilbert's 23rd problem. In the 20th century, it enjoyed the attention of Noether, Lebesgue, Hadamard and Carathéodory, among others.
All textbooks start from the same flagship examples, the catenary, the brachistochrone, and Dido's isoperimetric problem, and the same special cases, no explicit dependence on x, or no dependence on y. These are the "best" examples, the ones that arise from interesting problems and are at the same time "doable." They are also scarce resources, and van Brunt uses them wisely. He doesn't overuse them, but instead supports them with a rich assortment of lesser examples. The exercises have obviously been polished and sharpened in the classroom.
The subject itself is not so difficult, but the calculus of variations does have a substantial list of prerequisites, especially differential equations and classical mechanics. It would turn into a graduate level course if we also required normed spaces, but in exchange we would get to learn about the differences between weak and strong extremals and more about sufficient conditions. Van Brunt knows where that line is and stops where we can see what is beyond without crossing the line.
In total, this is a well crafted, reasonably priced book that would be a fine introduction to a fascinating subject that not enough mathematicians know about.[Ed Sandifer]
How Chinese Learn Mathematics
is a comprehensive look at nearly all aspects of mathematical development
in grades K-12 in Chinese educational systems.
The volume addresses all countries in which a significant amount of the population is Chinese. Issues discussed include, among others, comparisons between western and Chinese results in international tests, classroom techniques in the Chinese school system, comparisons between Chinese and western home environments for mathematical learning, and the historic connection to modern mathematics beliefs in Chinese educational systems. In depth perspectives regarding the use of textbooks, the recent movement to reform Chinese mathematics learning, and the preparation/beliefs of K-12 mathematics instructors in Chinese educational systems is presented.
This volume does a great deal to enlighten novices with regard to myths about mathematics learning in the Chinese system. For example, the myth that rote learning is the primary style used by Chinese students is addressed. The text does not deny that this is the type of learning most emphasized in the classroom. Instead, it elaborates on the specific details of the Chinese learning experience to show that "rote learning" is a loosely defined concept and that when scrutinized, much more than simple memorization and repetition exists in the Chinese mathematics learning structure.
The volume is well researched and referenced and the authors are generous in their criticism of what they believe to be weaknesses in Chinese mathematics learning systems.
This book has excited my interest in this topic. I strongly recommend it to anyone who is thinking of a career teaching K-12 mathematics, anyone who plans to prepare future teachers in mathematics, and anyone interested in mathematics curriculum development. I would, however, like to see more careful copyediting for English syntax in a few of the chapters. [Stephen Lancaster]
Publication Data
A
First Course in Abstract Algebra: Rings, Groups, and Fields, by Marlow
Anderson and Todd Feil. Chapman & Hall/CRC, 2005. Hardcover, 696 pp.,
$89.95. ISBN 1584885157.
Chance and Choice: Memorabilia, by Kai Lai Chung. World Scientific, 2004. Hardcover, 328 pp., $58.00. ISBN 981-256-012-2.
The Calculus of Variations, by Bruce van Brunt. Springer-Verlag, 2004. Hardcover, xiii + 290pp., $69.95. ISBN 0-387-40247-0.
How Chinese Learn Mathematics: Perspectives from Insiders, ed. by Fan Lianghuo, Wong Ngai-Ying, Cai Jinfa, and Li Shiqi. World Scientific, 2004. Hardcover, 592 pp., $88.00. ISBN 981-256-014-9.
Frederick M. Butler is Assistant Professor of Mathematics at the Institute for Mathematics Learning, West Virginia University.
Stacy G. Langton (langton@sandiego.edu) is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.
Ed Sandifer is a Professor of Mathematics at Western Connecticut State University in Danbury, Connecticut. His research interests are mostly historical, especially Leonhard Euler, and he has run the Boston Marathon every year since 1973. He can be reached at SandiferE@WCSU.edu.
Stephen Lancaster is a graduate student in Mathematics Education at the University of Oklahoma.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Sat Aug 27 14:09:45 Eastern Daylight Time 2005