A Conversational Introduction to Algebraic Number Theory

This book is a welcome new addition to the textbook literature on introductory algebraic number theory. It is clear, well-written and well motivated. The title promises a “conversational introduction”, and the book delivers: the author’s style is chatty and informal, with, for example, section headings with titles like “Goodbye, Norm” and “Will the real integers please stand up?”, and occasional jokes, such as his reference to the ring of integers as the “one ring to rule them all”.

But, make no mistake: its chatty tone notwithstanding, this is still a serious book, covering serious mathematics. Its prerequisites, for example, are not insubstantial. They include about a year’s worth of undergraduate abstract algebra (including at least some prior exposure to commutative rings, unique factorization domains and Galois theory), a prior course in linear algebra, and a decent acquaintance with elementary number theory, including quadratic reciprocity. For these reasons, I think the primary audience for this text would be graduate students (perhaps first year graduate students).

Although the author believes that many senior-level undergraduate math majors will meet these requirements, I am skeptical of this claim; in my experience, most senior-level undergraduates, except at elite institutions, lack the necessary background and mathematical maturity that would be necessary to really get a lot out of this text. However, as a graduate text, this book seems eminently suitable. It falls on the easier end of the spectrum, and seems more accessible than sources like Samuel’s Algebraic Theory of Numbers or John Milne’s excellent lecture notes. On the other hand, it omits some topics that an instructor of a graduate course might wish to include. For example, Samuel covers more sophisticated commutative algebra (Dedekind domains, for example) and topics like inertia groups. Milne’s notes contain chapters on valuations, and local and global fields. Jarvis’s Algebraic Number Theory, a somewhat easier text than this one, has a chapter on analytic methods, but this is also omitted in this text. Therefore, as a text for a graduate course, Pollack’s text might possibly require some supplementation.

The organization of this text is interesting. Recognizing that a lot of the theory of algebraic numbers can be motivated and made more accessible by limiting the discussion to quadratic number fields (see, for example, Trifkovic’s Algebraic Theory of Quadratic Numbers), Pollack spends the first 12 chapters of the book, roughly a third of the total text, dealing with quadratic extensions. Beginning with an expository chapter in which he explains how looking at algebraic numbers can shed light on problems in “ordinary” number theory, Pollack then proceeds to discuss, in the quadratic context, the major results of algebraic number theory. He defines the class number of a quadratic number ring and proves that it is finite, and also proves that although the ring itself may not enjoy unique factorization, there is unique factorization of ideals. The structure of the group of units of quadratic rings (both complex and real) is also discussed.

In addition to these fairly standard topics, Pollack also discusses some material that I thought was much less standard (and was, in fact, completely new to me). For example, although I have used, in several classes, the polynomial \(x^2+x+41\) as an example of a polynomial that produces many primes (for all nonnegative \(x\) less than 40, in fact) but which does not always produce prime values (no nonconstant polynomial can, of course; this one fails for the first time at \(x= 40\)), I never knew that there was a necessary and sufficient condition relating certain prime-generating polynomials to unique factorization of some quadratic number rings. This is the Frobenius-Rabinowitsch criterion, discussed in chapter 11.

After this introductory account via quadratic extensions, the book “reboots” and starts from scratch, revisiting the material in the more general context of arbitrary algebraic extensions of the rational numbers. My initial fear, when I read this description, was that it might result in a lot of duplication and wasted time, but that really did not turn out to be the case at all. The quadratic example motivates the general case and in some cases allows for simpler proofs, but the general case does add content. Overall, Pollack’s approach has the salutary effect of not only allowing the reader to learn the general theory but to see how the theory is affected by the more general assumptions. This itself adds insight and understanding to the development.

Each chapter ends with exercises (generally about six per chapter, give or take a couple), few if any of which struck me as particularly trivial. No solutions are provided, which I view as a pedagogical plus.

I do have two quibbles, one more serious than the other. The serious one concerns the total lack of a bibliography or list of suggestions for further reading. For a text aimed at upper-level undergraduates or early graduate students, this is disturbing. The omission is, I admit, ameliorated to some extent by the presence of numerous footnotes inserted throughout the text, but I do not believe these footnotes completely compensate for the omission of a selection of books and articles made available in one convenient place. And these footnotes do not, as far as I could tell, give the reader an idea of what other books on the subject are available.

The second quibble, which really boils down to a matter of individual taste, concerns what I thought was a missed opportunity. Given that the book begins with a discussion of how a mistaken assumption of unique factorization in a ring of algebraic integers can lead one astray when discussing the ordinary integers, it seems a pity to not at least mention one of the most famous examples of that fact, specifically Lamé’s faulty proof, announced in 1847, of Fermat’s Last Theorem. (A more expanded discussion of this can be found in the review of Learning Modern Algebra from Early Attempts to Prove Fermat’s Last Theorem by Cuoco and Rotman.)

To summarize and conclude: as far as undergraduate readers go, I continue to believe that Jarvis’s Algebraic Number Theory remains the most accessible source for this material. For graduate-level courses, this would be a very good choice as text, but some supplementation might be called for.

Buy Now

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.