You are here

Algebraic Number Theory

Frazer Jarvis
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Mark Hunacek
, on

For some time now, it has not been uncommon for undergraduate number theory texts to include at least some discussion of algebraic numbers. The well-known Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery has a chapter on the subject, as does the more recent An Introduction to Number Theory with Cryptography by Kraft and Washington. Stillwell’s Elements of Number Theory takes it a step further and heavily emphasizes the algebraic approach to the subject. Michael Artin’s Algebra also contains a chapter on quadratic number fields. So, undergraduate mathematics majors do have some convenient access to at least the most introductory parts of the subject. However, for fairly obvious reasons, this is usually as far as it goes for most of them; the average university does not generally offer a full-semester course on the subject, and undergraduate-level textbooks devoted entirely to algebraic number theory are not exactly thick on the ground.

This may be in the process of changing, however. Last November I reviewed Algebraic Theory of Quadratic Numbers by Trifković, a book which aimed to make the subject accessible to undergraduates by focusing on quadratic fields, where ad-hoc methods could replace more generally applicable but more sophisticated methods. And now, not a year later, we have the book under review, which is part of Springer’s excellent Student Undergraduate Mathematics Series (SUMS).

There is substantial, but by no means total, overlap between the topics covered in both books. Jarvis, like Trifković, recognizes the value of discussing quadratic extensions in detail, but, unlike Trifković, also ventures beyond quadratics. Chapter 1 begins with a review of basic divisibility properties of the integers, but also introduces the Gaussian integers. This serves several purposes: first, it shows how information about the integers can be obtained by looking at larger systems, and it also provides another example of unique factorization, thereby helping bringing this topic to the forefront of discussion.

Algebraic and transcendental numbers are introduced in the next chapter, and the existence of the latter established by both constructive (Liouville’s approximation theorem) and non-constructive (comparison of cardinalities of the sets of real and algebraic numbers) methods. The author develops, as necessary, the basics of the elementary theory of field extensions (minimal polynomials, etc.); this in fact turns out to be a common thread running through the book: in an effort to make this text generally accessible to undergraduates, quite a lot of it is devoted to introductory accounts of the kind of abstract algebra that might generally be taught in a second course in the subject (Noetherian rings, PIDs, Euclidean domains, etc.).

Having explained what the minimal polynomial is, the next natural step is to introduce the notion of algebraic integers. The author does so, and the ring of integers of the quadratic field \(\mathbb{Q}(\sqrt{d})\) for arbitrary square-free \(d\) is worked out in detail. The next chapter continues the study of rings of integers by introducing some basic properties: norm, trace, integral basis and discriminants. With this machinery developed, the integers of some non-quadratic algebraic number fields are also worked out.

There follow two chapters that discuss the theme of unique factorization, which is central to any study of algebraic number theory. After illustrating non-unique factorization in some quadratic fields, it is explained how Kummer tried to remedy this phenomenon by introducing the concept of “ideal number”, which in turn led to Dedekind’s definition of “ideal”. After developing some commutative algebra, unique factorization into prime ideals is established for the ring of integers of an algebraic number field. Fractional ideals and the class group are also defined.

The next chapter (chapter 6) consists largely of detailed examination of specific examples — specifically, quadratic imaginary number fields. (The author states at the beginning of the chapter that real quadratic fields are somewhat different and will be studied later.) The author first proves a theorem identifying all negative squarefree integers d (there are only five of them) with the property that the ring of integers of \(\mathbb{Q}(\sqrt{d})\) is Euclidean. He then introduces connections with quadratic forms to investigate the much more difficult question of which values of \(d\) give rings of integers with unique factorization. Though the full result here is beyond the scope of the text, the author does show how quadratic forms can be used to make partial progress on the problem.

Chapter 7 introduces geometric reasoning (via the concept of a lattice) into the theory. Minkowski’s theorem is stated and proved, and then applied to obtain as consequences the finiteness of the class number and Dirichlet’s Unit Theorem.

The next two chapters apply all this theory to study several examples, including real quadratic algebraic number fields (for which a detour into the basic facts of continued fractions is provided), several cubic and quartic examples, and cyclotomic fields. As an application of cyclotomic fields, the author provides a discussion of Gauss sums and a proof, using them, of the law of quadratic reciprocity. Connections with Fermat’s Last Theorem (FLT) are also discussed. (One mild complaint here: I think that the failure to mention Lamé and his mistaken belief that he had proved FLT can only be described as a lost opportunity.)

The final two chapters of the text cover more advanced topics that are outside the mainstream of a typical undergraduate course and, I would guess, would almost never be reached in a course based on this book (but of course one cannot fault the author for including them and therefore making the text a bit more versatile): analytic techniques (including a discussion of the Riemann zeta function, the zeta function of an algebraic number field and the analytic class number formula), and the number field sieve, a relatively new (about 25 years old) factorization algorithm (the fastest one known, according to the author) which uses algebraic numbers. This chapter also contains, by way of introduction and explanation of why anybody would care about factorization, a brief discussion of the RSA cryptosystem. (For much more on factorization, see The Joy of Factoring by Wagstaff.)

These topics conform more closely to those that I think should be considered in a first course on algebraic number theory than do the ones covered in Trifković, particularly in one major respect. I firmly believe that all undergraduate mathematics majors should learn something about FLT before graduating, and one natural place to be exposed to this topic is in a course on algebraic number theory. (See, e.g., Learning Modern Algebra from Early Attempts to Prove Fermat’s Last Theorem by Cuoco and Rotman.) The connection, as previously noted, appears via cyclotomic fields; since Jarvis discusses them and Trifković does not, it is not surprising that FLT is discussed in the former but not the latter. (Readers who want even more of a discussion of FLT in a reasonably introductory text might wish to look at Algebraic Number Theory and Fermat’s Last Theorem by Stewart and Tall, which unlike this text contains a lot of material on elliptic curves. I doubt that Stewart and Tall is really suitable for an undergraduate audience, however. Readers of that book should note that theorem 13.20, which characterizes the groups that can possibly appear as the torsion subgroups of an elliptic curve, is incorrectly stated: \(\mathbb{Z}/12\mathbb{Z}\) is one such possibility that is not mentioned.)

I thought highly of the Trifković book, but the selection of topics in Jarvis’s text is one reason why, if anything, I like the latter a bit more. In addition, my (admittedly subjective) impression is that this book is a bit easier and more reader-friendly than Trifković’s, and therefore perhaps more suitable for an undergraduate audience. (Trifković’s book appears in the Universitext series, one that is not intended solely or even primarily for undergraduates.)

The text contains lots of exercises, just about all of which have solutions appearing in a thirty-page long Appendix. The availability of solutions is typical of books in the SUMS series, and has both advantages (making the book more accessible for self-study) and disadvantages (for obvious pedagogical reasons). My general view is that the disadvantages predominate, and I am typically somewhat reluctant to select texts where all the exercises have solutions. However, since the subject matter of this text is not generally covered in standard undergraduate courses and the book is likely to be used for seminars or special reading courses with well-motivated students, the presence of these solutions may not be as problematic as in lower-level texts.

Lawrence Corwin, late of Rutgers University, once amusingly wrote that Serge Lang’s texts “often have the property of making every other book in the field seem like a pedagogical advance.” Lang did author a book on Algebraic Number Theory (though certainly not an undergraduate level one), but Jarvis’s book would appear to be a pedagogical advance even if Lang’s book had not been written. It is, I think, the clearest and most accessible account of these topics currently available. I recommend it highly.

Mark Hunacek ( teaches mathematics at Iowa State University.