Read This!

The MAA Online book review column

Briefly Noted

April 2005

This text instructs students in solving and using differential equations with both paper-and-pencil techniques and the Mathematica symbolic manipulation program. Mathematica commands assist the solution of the differential equations, streamlining tedious computations while paralleling the paper-and-pencil steps. Ross acquaints the reader with Mathematica commands by example with a minimum of syntax explanation. Mathematica use is interactive, avoiding programming and user-created multi-step commands. Occasionally the text introduces "black-box" Mathematica commands which solve differential equations directly, but this typically just mentions such commands exist and checks or compares the solutions to the lengthier step-by-step computation. Mathematica also functions in this book as a way for students to check answers and in fact the book does not have solutions to problems in the back. Instead, Ross explicitly directs students to check their answers by modifying the Mathematica examples. The book comes with a CD with notebooks providing further examples of using Mathematica.

The differential equations topics covered are the usual sequence of first-order equations, standard applications of first-order equations, higher-order linear constant coefficient equations, applications to oscillations and electrical circuits, Laplace transform methods, series solutions and linear systems with phase-plane diagrams. A quick survey of linear algebra appears in Chapter 2, and then the language of linear algebra describes the solution space of differential equations throughout the book. Second-order and higher-order equations are in the same chapter in subsequent sections. The techniques of variation of parameters and undetermined coefficients are treated equally for non-homogeneous equations, and undetermined coefficients is approached through the method of annihilators.

In spite of the availability of Mathematica, numerical and graphical methods are not emphasized. Direction fields and Euler's method appear in the first chapter but do not reappear in the chapter on applications of first-order equations. The fourth-order Runge-Kutta algorithm is mentioned as a higher-order extension of Euler's method along with Mathematica's NDSolve command. In the chapter on applications of first-order equations, the only numerical applications are a numerical integration for a specific solution and NDSolve is used for a Ricatti equation. In two very short sections in the chapters on applications of second-order equations and on systems, Ross derives a nonlinear equation for planetary motion and mentions the Lotka-Volterra equations, but neither is explicitly solved numerically. Plotting solutions with Mathematica is not prominent, for example, in the chapter on first-order equations the text only demonstrates plotting a family of solutions of a Bernoulli equation, and plotting the solutions of the Clairaut equation. Plotting of solutions is more prominently displayed in the section on damped second-order equations. No Mathematica plotting commands are in the chapter on systems, although the many figures have obviously been prepared with Mathematica. I did not receive the accompanying CD for review, maybe the examples I am looking for are there.

The index does not include Mathematica commands. There are several places in the book where bad typography mangled some mathematical symbols (for example rendering the ≠ sign into "\ ="), and this could puzzle some students.

So what do we have here? This is a good, thorough standard textbook with good explanations. However, this textbook is very similar to the textbook I learned elementary differential equations from, with the addition of Mathematica commands to automate calculation and further illustrate the solution techniques. I have seen the future, and it works just like the past. [Steven R. Dunbar]

Some courses in mathematics have a single textbook that has become an almost canonical choice — Rudin's analysis book, Munkres' Topology book, and Ahlfors' complex analysis are extremely common choices as one flips through syllabi. There does not seem to be any such canonical choice for an abstract algebra textbook, though there are several contenders for that title. One of these books, Abstract Algebra by David Dummit and Richard Foote, has just released a third edition.

The topics covered in this book are, if not completely standard, also certainly nothing surprising. The first six chapters deal with groups: basic definitions, examples, the Sylow theorems, group actions, and a variety of related topics. The next three chapters define rings, and discuss the basic properties of rings and domains. One of these chapters is devoted exclusively to polynomial rings — and in this chapter the authors introduce Grobner bases, the topic which really distinguishes this edition from previous ones. The next three chapters deal with modules and vector spaces before moving on to a chapter on field theory and a separate complete chapter on Galois theory.

The last four chapters consist of topics which I would consider beyond the scope of most one year courses in algebra — commutative rings, discrete valuation rings, Dedekind domains, and a bit of algebraic geometry, in addition to nice introductions to homological algebra, group cohomology, and the representation theory of finite groups. An appendix contains some category theory, enough to placate those users who want to learn the material, while not enough to distract from the main thrust of the book. In the introduction, the authors give several suggestions as to what material one might include in a one-year course, though one of the appeals of a book such as this one is that the professor can pick and choose from a wide variety of topics.

As mentioned above, the main addition to this edition of the book is the introduction of several sections on Grobner bases. The authors revisit the topic several times after initially introducing the concepts, and unlike many later editions of books, these additions flow quite well with the rest of the material. In particular, the exposition and the exercises are of as high a quality as one would expect from a textbook that is as ubiquitous as this one. Certainly there are other Algebra books that might be more appropriate for some audiences — this book is far from the easiest book on the market, and also far from the most technical — and there are topics that I think are presented better in Gallian or Hungerford — but there is no doubt that this book is one of the best on the market in what it does, and anyone looking for an algebra textbook should give Dummit and Foote a serious look. [Darren Glass]

Most professors would agree that a student who truly wants to learn any given area of mathematics will benefit enormously from solving many problems on the subject at hand. Struggling with a problem often helps one understand the finest aspects of the theory. However, can a student learn an area of mathematics just by solving a systematic collection of problems?

Problems in Algebraic Number Theory is intended to be used by the student for independent study of the subject. It provides the reader with a large collection of problems (about 500), at the level of a first course on the algebraic theory of numbers (with undergraduate algebra as a prerequisite). The volume also includes completely spelled-out solutions to all exercises. The list of topics include elementary number theory, algebraic numbers and number fields, Dedekind domains, ideal class groups, structure of the unit group, reciprocity laws (quadratic and higher) and Dirichlet L-functions. A new chapter on density theorems was added in the second edition. Each section starts with some basic definitions and theorems (with proofs), and a couple of solved examples.

The reviewer thinks that the authors have done a fantastic job choosing the problems, which are perfectly arranged so the students can progressively move on from topic to topic, discovering on their own the proofs of the most well-known results and applications. The exposition of the solutions is very clear and helps to introduce different important techniques.

However, the reviewer believes that a student is better served with a healthy balance between traditional lectures and problem-solving. Without proper advising the student might miss the 'soul' of the subject. The book feels at times like a bare sequence of definitions, theorems and exercises. The readers are supposed to find out, on their own through problem solving, why the concepts are introduced, and this might be a path that not everybody can easily follow. The student might find hard to differentiate routine problems from the more relevant exercises which will be used as lemmas in consequent sections. It is the opinion of the reviewer that each chapter should be complemented with a couple of lectures by a professor to emphasize the main ideas and clarify the goals of the developing theory. In any case, the book is an excellent resource for the instructor and the student as a companion to any algebraic number theory course. [Álvaro Lozano-Robledo]

Bateman and Diamond's Analytic Number Theory is a graduate level textbook in the subject. The primary prerequisites are beginning graduate level courses in complex analysis and real analysis. A prior course in elementary number theory would be useful but not necessary.

The text covers standard topics such as the distribution of primes numbers, including Dirichlet's theorem on primes in arithmetic progressions, and basic properties of the Riemann zeta-function and Dirichlet L-functions. The authors give both elementary and analytic proofs of the Prime Number Theorem. A less standard, but very interesting topic, is a thorough treatment of Dirichlet convolution, including a discussion of convergence of infinite convolution products. The authors present the Weiner-Ikehara approach to the Prime Number Theorem, along with generalizations that lead to the asymptotic formula for number of sums of two squares.

The book also includes a nice introduction to sieve methods. The Brun-Hooley sieve is presented in the general formulation developed by the author's colleagues K. Ford and H. Halberstam. This sieve is "revisionist history"; this is not a sieve that Brun developed, but it is a nice variation that Hooley found that is very much in the spirit of Brun's approach. The authors also present the large sieve using the Beurling-Selberg extremal function approach. Finally, the authors present some applications of sieves, including the Brun-Titchmarsh upper bound for primes in intervals and upper bounds for the number of twin primes.

Overall, this is a nice well-written book with plenty of material for a one-year graduate course. It would also make nice supplementary reading for a student or researcher learning the subject. [S. W. Graham]

Publication Data

Abstract Algebra, Third Edition, by David S. Dummit and Richard M. Foote. Wiley, 2003. Hardcover, 944 pp., $119.95. ISBN 0-471-43334-9.

Problems in Algebraic Number Theory, Second Editon, by Jody Esmonde and M. Ram Murty. Graduate Texts in Mathematics 190. Springer-Verlag, 2005. Hardcover, xvi + 352 pp., $59.95. ISBN 0-387-22182-4.

Analytic Number Theory: An Introductory Course, by Paul T. Bateman and Harold G. Diamond. World Scientific, 2004. Hardover, 376 pp., $78.00. ISBN 981-238-938-5. Paperback, 376 pp., $42.00. ISBN 981-256-080-7.

Steven R. Dunbar (sdunbar@unl.edu) teaches at the University of Nebraska - Lincoln and is MAA Director of Mathematics Competitions.

Darren Glass is a VIGRE Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at glass@math.columbia.edu.

Álvaro Lozano-Robledo is Visiting Assistant Professor of Mathematics at Colby College. He can be reached at alozano@colby.edu.

S. W. Graham is currently a Professor of Mathematics at Central Michigan University. He can be reached at sidney.w.graham@cmich.edu.

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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.