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Algebraic Theory of Quadratic Numbers

Mark Trifković
Publication Date: 
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[Reviewed by
Mark Hunacek
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Sometimes, the basic ideas of a subject that is not typically considered undergraduate fare can be made comprehensible to undergraduates by focusing on concrete special cases that are nevertheless broad enough to convey the flavor of the material. Thus, for example, a substantial chunk of the theory of Lie groups (whose very definition requires sophisticated mathematical ideas like manifolds) can be taught to undergraduates by restricting attention to one particular kind of Lie groups: namely, matrix groups. (See, for example, Stillwell’s Naive Lie Theory.) By restricting attention to matrix groups and using arguments that work for them but do not work in general, much of the flavor of the theory can be discussed at the undergraduate level. For example, the exponential map in this context can be defined explicitly as a matrix power series, and its basic properties proved. Such an approach, it seems to me, is a splendid idea, since it not only gives undergraduates a taste of mathematics that used to be considered beyond their reach, but also gives them, when they eventually learn the more general theory, an excellent source of examples and intuition.

The book under review can roughly be described as doing for the theory of algebraic numbers what Stillwell’s book does for Lie groups. Typically, algebraic number theory is the subject of books, such as Marcus’ Number Fields or Lang’s Algebraic Number Theory, that are intended for a graduate-level audience and, accordingly require a fairly sophisticated background in algebra. Marcus’ book, which I think is generally regarded as being one of the more accessible ones, states in the preface that the first six chapters can be read by anybody who is familiar with “the most basic material covered in standard undergraduate courses in linear algebra and abstract algebra” but then states in the next paragraph that chapter 4 assumes knowledge of Galois theory.

The central idea of this book is to focus attention solely on quadratic number fields and their rings of integers, thereby allowing the use of ad hoc techniques that may not generalize, and thus reducing the prerequisites to, in the author’s words, “a knowledge of elementary number theory and a passing familiarity with ring theory”. Even these topics are reviewed in the text: the first section of chapter 1 summarizes the basics of elementary number theory and chapter 2, “A Crash Course in Ring Theory”, reminds the reader of the basics of that area: homomorphisms, ideals, quotient rings, prime and maximal ideals. In particular, Galois theory is not required here. In fact, this book was designed, the author tells us, “to give the undergraduate a taste of algebraic number theory right after (or during) the first course in abstract algebra”.

Taken literally, this seems like an overstatement: I don’t imagine that much of this book would go down very well during a first course in abstract algebra, where students with minimal algebraic maturity are typically struggling to come to grips with new concepts like groups and rings. I also don’t think, at least based on personal experience, that most instructors of such a first course would have time to do the kinds of things that are presented here. However, to the extent that the text intends to bring the subject of algebraic number theory down to the undergraduate level, it succeeds admirably: it is written at a level that should be comprehensible to good undergraduates, contains many examples, and has a number of exercises (solutions to some, but not all, of which appear in a short three-page Appendix).

Chapter 1, in addition to summarizing some basic number theory in the ring of integers, also introduces a few other specific examples of quadratic number fields and their associated rings of integers. We see the ring of Gauss integers (the author prefers that term to “Gaussian”) and the ring of Eisenstein integers, both of which have unique factorization, and two other examples, one real and one imaginary, which do not. A first look is given at unique factorization of ideals, a subject explored in more depth subsequently.

After the previously-mentioned chapter 2 (reviewing ring theory), chapter 3 discusses lattices and column reduction of integer matrices, a tool that is used to study lattices and also to facilitate calculation with ideals. This leads to chapters 4 and 5, which develop most of the machinery of a typical introductory algebraic number theory course, in the specific context of quadratic extensions. All the usual suspects make an appearance here, including: trace and norm; the ring of integers of a general quadratic field (defined in a somewhat nonstandard way in terms of trace and norm); Noetherian rings (called “Noether rings” here; the author seems to dislike words ending in “ian”); fractional ideals; inert, ramified and split prime ideals; and the ideal class group and class number. Unique factorization of ideals, referred to in chapter 1, is proved, and quite a few class number calculations are provided.

The next chapter develops the theory of continued fractions and applies that theory to study the units of the ring of integers of a real quadratic number field. The methods used are a good example of what was referred to earlier: ideas that work nicely in quadratic extensions but do not readily generalize to the arbitrary case. The same can be said for the final chapter of the book, which defines quadratic forms and applies them to quadratic extensions. This is the longest, most technical and most difficult chapter of the book, and will undoubtedly present the most challenges for an undergraduate reader; however, even if it is skipped entirely, the first six chapters should comprise a good semester’s course.

Of course, the idea of studying quadratic extensions as a way of learning the basics of algebraic number theory is not new to this author (and he certainly does not claim that it is): the classic text Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery has been around for decades now, and contains a chapter on algebraic numbers which focuses on quadratic extensions as a way of illustrating the general theory, as does the much more recent second edition of Saul Stahl’s Introductory Modern Algebra, a historically-themed introduction to abstract algebra. These are only chapters, but a substantial portion of John Stillwell’s Elements of Number Theory is also devoted to quadratic extensions, and the same is true of Cohn’s Advanced Number Theory.

I think, however, that this book is sufficiently different from these other references so as to be considered a fairly novel addition to the existing textbook literature. In terms of difficulty and level of coverage, it lies between Stillwell and Cohn: it is more demanding than, and contains material not found in, Stillwell (which is not really intended as a text for a course on algebraic number theory, and which covers many topics traditionally found in a first course on elementary number theory, even down to the definition of “a divides b”) while at the same time is considerably more reader-friendly and less advanced than Cohn, which contains a lot of material not found here (such as Dirichlet characters, L-series and Dirichlet’s theorem on primes in arithmetic progression). Cohn’s book is also more than fifty years old; the current Dover edition is a reprint of a book first published in 1962 under the title A Second Course in Number Theory.

To summarize: this is an interesting and well-written book. It would be interesting to see if it really does result in more undergraduates learning about algebraic number theory; based on the quality of this text, that seems like a very real possibility.

Mark Hunacek ( teaches mathematics at Iowa State University.