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Algebra and Number Theory: A Selection of Highlights

Benjamin Fine, Anthony Gaglione, Anja Modenhauer, Gerhard Rosenberger, and Dennis Spellman
Walter de Gruyter
Publication Date: 
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De Gruyter Textbook
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
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Teaching an upper-level mathematics class can be a lot of fun, but can also be frustrating in some respects. It seems, for example, that in every such course I’ve taught, there have been some interesting side topics or alternative points of view that I really wanted to talk about in class, but, for lack of time, simply couldn’t get to. For example, in an elementary number theory course, isn’t it obvious that the “right way” to think about Fermat’s Little Theorem is as a consequence of Lagrange’s theorem, applied to the group of nonzero residue classes, modulo a prime p, under the operation of multiplication? But in a beginning course, populated by students many of whom have not taken a course in abstract algebra, this kind of detour takes time that just isn’t generally available.

The book under review wants to do something about this. Addressed to students who have already had some exposure to abstract and linear algebra (the book does define certain terms, such as “group” and “ring”, but it seems clear that these definitions are intended to jog the memory rather than teach these ideas for the first time), this book discusses a number of what the authors refer to as “gems” of algebra and number theory, and also offers some indication of how these two subjects complement and enhance each other.

The first five chapters are largely number-theoretic in nature. They cover the basic facts of elementary number theory (natural numbers and integers, divisibility and primes, congruences, Pythagorean triples and sums of squares). Basic group and ring theory is used throughout. After these chapters, there are several on topics in algebra, specifically related to polynomials (including symmetric polynomials) and field extensions. The scene then shifts back to numbers — the real numbers, complex numbers, p-adic numbers, quadratic field extensions, transcendental numbers, and constructible numbers. Algebraic methods are again used to enhance these chapters. The book then ends with a chapter on geometry from the point of view of real inner product spaces.

Along the way, a lot of interesting topics and results are collected and discussed. These include:

  • The Peano axioms for the natural numbers
  • Multiple proofs of the infinitude of prime integers — the standard proof of Euclid, and also proofs using Fermat numbers and Fibonacci numbers
  • Pythagorean triples, characterized using the chord technique
  • Fermat’s theorem on sums of two squares, proved using the projective special linear group
  • Horner’s method for evaluating polynomials at a point
  • The field of real numbers, constructed using Cauchy sequences of rational numbers
  • Dirichlet’s theorem on approximating irrational numbers by rational ones
  • The field of p-adic numbers
  • Several different proofs of the Fundamental Theorem of Algebra
  • Cardano’s formula for the solution of a cubic polynomial
  • A direct proof, similar to Abel’s, of the insolvability of the quintic
  • Discussion of the calculation of \(e\) and \(\pi\)
  • Proof of the irrationality of \(e\) and \(\pi\)
  • Proof of the transcendence of \(e\) and \(\pi\)
  • Proof of the transcendence of Liouville’s constant \(\lambda\)
  • The Wallis product formula
  • Some discussion of the Riemann zeta-function
  • Proofs of the impossibility of the famous Euclidean construction problems
  • A statement and proof of the necessary and sufficient condition on n for constructibility of a regular n-gon

I am not aware of any one other book that covers all of the material described above. (One that comes close is Abstract Algebra by Carstensen, Fine and Rosenburger, but that’s sort of cheating, since two of the three authors of that book are also co-authors of this one.) For this reason, the book under review seems like a very useful reference, one that anyone with an interest in algebra or number theory would enjoy having readily available.

The book’s suitability as an actual text for a course, however, is an entirely different question. This book likely suffers from something of an audience problem. There are not many courses offered that cover this array of topics, so the audience is, from the outset, somewhat limited. For example, although the first five chapters of this text intersect nontrivially with a standard undergraduate textbook in number theory, most such standard texts do not presuppose the level of algebraic sophistication that this book does. Also, instructors of a typical number theory course might want to cover some topics (e.g., quadratic reciprocity) that appear in most number theory texts, but which are not found here.

Many universities offer a second course in abstract algebra, but here too there are problems with the use of this book as a text for such a course. For one thing, the emphasis on number theory might not be appropriate for such a course, and as before there are also topics that one might want to cover in such a course (e.g., Sylow theory or Galois theory) that are not addressed in this book.

The authors state that they see the book as being primarily used for Masters-level courses for future teachers; the problem is that, to my knowledge anyway, not many American universities offer such courses. (Iowa State University, for example, used to have a Masters of School Mathematics program, but it no longer exists.)

Limiting the prospective audience still further is the authors’ writing style, which is rather terse and concise. Things are occasionally stated in a fairly precise way, but without much preliminary motivation. For example, a polynomial with coefficients from an integral domain R is initially defined as a function f, defined on the set of nonnegative integers and mapping into R, with the property that f(i) = 0 for all but finitely many i. Only after this definition is given, is it pointed out that f may be thought of an infinite sequence of elements of R, all of them 0 from some point on, and only after this observation is made is the connection with “formal expressions” given explicitly.

Occasionally, also, specific definitions are omitted. The authors point out, for example, that any field contains a smallest subfield, but neglect to explicitly mention that this is called the prime subfield of the field, even though that term is then used in a theorem immediately following. Also, in defining the minimal polynomial of an algebraic element, the authors refer to a polynomial as “normed” without explaining the meaning of that word; I assume they mean the polynomial is normalized to have leading term 1, which makes one wonder why they just didn’t call it “monic”, a term that they have previously defined.

In addition, the language is often somewhat stilted, as though badly translated. Definite articles like “the” or “a” are frequently omitted (“The polynomial … is called minimal polynomial of a”; “L is called field extension of K”) and, more importantly, sentences can be syntactically confusing. (For example: “In mathematics, when developing a concept or a theory, it is often not possible, all used terms, properties or claims to prove, especially existence of some mathematical fundamentals.”)

And, as long as we’re discussing English usage, the authors refer to the famous Delian problem of doubling the cube as the “problem from Deli”, suggesting that it originated in a place called Deli rather than the island of Delos. (Reading this, I had an amusing mental image of getting a compass and straightedge with my pastrami on rye.)

The publisher’s webpage for this book states that “[t]his two-volume set collects and presents…”, but it should perhaps be clarified that this book is a single, stand-alone, volume. My understanding is that a second book, as yet unpublished and unadvertised on the De Gruyter webpage, will also be a “selection of highlights” text, this one in the areas of geometry and discrete mathematics. The book now under review is sufficiently interesting that I am looking forward to this second volume. I think, however, that this is a book that is more likely to be enjoyed by faculty members than by students, who might find it somewhat difficult.

Mark Hunacek ( teaches mathematics at Iowa State University. 

See the table of contents in the publisher's webpage.