Introduction to Number Theory

This short (about 200 pages of actual text) book is intended as a text for an introductory course in number theory at the (British, anyway) undergraduate level. It has several features that distinguish it from its competition, but I suspect that for most instructors, these features will militate against, rather than in favor of, its adoption as a text, at least at American universities.

Probably its most significant distinguishing feature is that this book is more algebraically oriented than most undergraduate number theory texts. It assumes prior knowledge of basic group theory (including familiarity with Lagrange’s theorem on the order of a subgroup of a finite group) and also some acquaintance with the notion of a field. Throughout, the subject is approached from an algebraic statement. The book begins, for example, with a discussion of the ring structure of the set of integers modulo \(n\).

Anybody who asserts that many of the theorems of basic number theory are best understood as special cases of results in algebra (for example, Fermat’s Little Theorem is an immediate consequence of Lagrange’s Theorem, applied to the multiplicative group of nonzero residue classes modulo a prime \(p\)) will get no argument from me. The problem is, many (likely most) number theory courses at American universities do not require abstract algebra as a prerequisite, and also draw students from disciplines other than mathematics, the most obvious example being computer science. The use of ad hoc techniques, rather than resort to general algebraic theories, can be frustrating for an instructor of such a course, but things are what they are, and the reality is that most instructors of basic number courses cannot assume that students already have this algebraic background, and also do not have the time to develop it as needed.

There are some books — Stillwell’s Elements of Number Theory being an excellent example — that develop the algebra as necessary and use it to prove things like Fermat’s Little Theorem, and when I taught number theory several years ago I gave some thought to using this book, but ultimately concluded it would be just a bit too difficult for most of my prospective audience. I certainly would not have even considered a book that, like the one now under review, does not develop basic group theory but instead assumes it.

It should also be noted that, at times (especially in chapter 4), a reader of this book is assumed to know some basic real analysis, at least to the point where the idea of a convergent sequence is not something totally foreign.

As mentioned, chapter 1 of this book begins with the ring structure of \(\mathbb{Z}/(n)\) and ultimately proves that this is a field when and only when \(n\) is a prime number. After a brief discussion of rings, the author rapidly discusses the basic facts about divisibility; interestingly, he does not prove the Division Algorithm (the statement that one integer can be divided by another to produce a suitable quotient and remainder); instead, he apparently views this as an obvious fact about the integers. (I don’t much care for that approach; I think this result is something students should see proven, as a consequence of other basic facts about the integers.) The greatest common divisor (the text calls it the highest common factor) of two integers is discussed and proved to be a linear combination of the two integers, and the basic facts about primes are established as well. One nice thing about this chapter is that the author takes pains to point out that uniqueness of factorization into primes is a more subtle idea than many might at first think, and gives a quick example (\(\mathbb{Z}[\sqrt{-5}]\)) to illustrate that it need not hold in all rings.

The next chapter seems to be more algebra than number theory. It discusses polynomials, including such things as the Eisenstein criterion, and defines the terms “unique factorization domain” and “Euclidean domains”, relating these ideas to the ring of polynomials over a field.

The third chapter combines the themes of the first two and discusses congruence equations modulo a prime. A high point is the law of quadratic reciprocity, which is stated in one section and proved (using Gauss sums) a few sections later.

This leaves two chapters, each independent of the other. Chapter 4 discusses \(p\)-adic numbers and some applications, motivated by the problem of solving congruences modulo prime powers. This is material that is not often covered in elementary number theory texts, and there may be a reason for that; although the author does a good job of trying to make this material accessible, I had the sense that this chapter would not work well in a typical undergraduate course in this country.

Chapter 5, following up on the brief example given in chapter 1, discusses quadratic integer extensions and the possible lack of unique factorization. This chapter (which seemed to me to be more accessible than the previous one) can serve as a very brief introduction to some of the ideas in algebraic number theory (but not ideals; the author stops short of introducing these) and also discusses some applications of these ideas to “ordinary” number theory (e.g., Diophantine equations and continued fractions).

Just as a book can be distinguished from others by the topics that it covers, it can also be distinguished by the topics that it omits. I was surprised to see no mention of Fermat’s Last Theorem, for example; Hill refers to Fermat’s Little Theorem but says nothing about what makes it “little”. I don’t recall seeing Goldbach’s conjecture stated, or perfect numbers defined. Pythagorean triples are also not discussed. Neither are sums of squares (two, three or four). Applications to cryptography are mentioned (a brief section introduces RSA and Diffie-Hellman), but not discussed in the kind of depth that student interest would seem to warrant.

The omission of any discussion of Fermat’s Last Theorem (FLT) seems like a particularly egregious missed opportunity, not only because of the significance of this result in number theory but also because the first section of chapter 5 is devoted to Diophantine equations and unique factorization. In view of the fact that a famous faulty proof of FLT was once proposed based on an incorrect assumption of unique factorization in certain domains (see, for example, Learning Modern Algebra from Early Attempts to Prove Fermat’s Last Theorem by Cuoco and Rotman), this section would certainly have been a good place to at least mention FLT.

To summarize and conclude: Since I teach American undergraduates at an American university, I can only express an opinion as to the suitability of this book for a course at such a university. For the reasons stated above, I don’t think this text would work very well in this context: it demands too much in the way of background, covers topics that might be viewed as too sophisticated for a first course, and omits some topics that many instructors would want to cover. At the same time, it also seems unsuitable as a text for a graduate course in number theory, because it also omits most of the topics that one would want to cover in such a course: there are no proofs of the Prime Number Theorem, Dirichlet’s theorem, etc. An instructor of an undergraduate course with very well prepared and sophisticated students might find it valuable, however. Graduate students who want a quick introduction to \(p\)-adic methods and quadratic rings might also benefit from a look at chapters 3 and 4.

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