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Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

J. L. Lehman
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions
[Reviewed by
Fernando Q. Gouvêa
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Algebraic number theory has never really been part of the “common core” of mathematics. When it was created in the late 19th century, it was considered arcane and abstract. Today, it is taught mostly at the graduate level, and even then mostly to students whose interests are in closely related topics. It’s perfectly possible to get a Ph.D. in mathematics without ever learning about algebraic numbers, ideals, or class groups. So it is something of a reach to attempt to write an undergraduate textbook on the subject.
One of the difficulties has to do with pre-requisites. A full-scale account of the subject requires knowing not only about linear algebra, groups, rings, and fields, but also about Galois theory, lattices, valuations and the topologies they determine, complex analysis, and probably more. All undergraduate accounts have to find a way around this. Mark Hunacek has discussed some of the options in his reviews of books by Frazer Jarvis and Paul Pollack.
Lehman’s approach is signaled by his title: he focuses only on quadratic number fields and the quadratic forms that are closely related to them. The fields in question look like \(\mathbb Q(\sqrt{D})\) where \(D\in\mathbb Z\) is squarefree. Connected to them are binary quadratic forms \(ax^2+bxy+cy^2\) such that the squarefree part of the discriminant \(b^2-4ac\) is \(D\). Both theories include a notion of equivalence and a ”class group“ formed by the equivalence classes.
The theory of quadratic forms goes back to Gauss’s Disquisitiones Arithmeticae, which of course does not use the language of number fields. This theory was the heart of Dirichlet’s Lectures on Number Theory. It was in an appendix to this book (not, alas, included in the translation) that Dedekind first introduced his theory of ideals, with the aim of giving a simpler account of Gauss’s theory of the composition of forms. Lehman here reverses the historical order, starting with quadratic number fields and then going on to develop the theory of quadratic forms and detailing how the two theories are related.
There are several advantages to focusing on the quadratic case. One can avoid Galois theory, for example, by introducing only the automorphism that sends \(\sqrt{D}\) to \(-\sqrt{D}\). It is possible to determine the ring of algebraic integers explicitly and to check directly that it is a ring. The \(\mathbb Z\)-modules being studied are all of rank two. All sorts of difficult arguments can be made simpler.
So much has been done before, for example in Algebraic Theory of Quadratic Numbers by Mark Trifković and Advanced Number Theory by Harvey Cohn. But Lehman adds a twist or two. He spends a lot more time on the theory of quadratic forms than usual. More significantly, he focuses on computational control of the material: he wants his students to learn how to compute with ideals in quadratic fields, to be able to compose two quadratic forms, and even, in some cases, to completely determine class groups.
To do this, Lehman uses some special properties of quadratic fields. First, one can find and fix an explicit integral basis \(\{1,z\}\) for the ring of integers of the field \(\mathbb Q(\sqrt D)\). Next, any ideal in that ring of integers (indeed, in any order in that field) can be written in the special form \[ I = \{g\left(ma+n(k+z)\right)\mid n,m\in\mathbb Z\},\] where \(g\), \(a\), and \(k\) are integers and \(k\) is determined modulo \(a\). Lehman writes this ideal as \(g[a:k]\). Indeed, he reverses the process by defining “ideal numbers” as symbols of this kind subject to certain conditions on the integers \(a\) and \(k\), and only later defining ideals. Throughout the book, algorithms are found for doing computations with ideals in this canonical form. (When are two such ideals the same? When are they equivalent? How can we mutliply two of them?)
(For experts, here's what is going on. Fix the canonical integral basis \(\{1,z\}\) of an order \(\mathcal O\). Then any ideal in \(\mathcal O\) is the \(\mathbb Z\)-span of two elements of \(\mathcal O\), \(a+cz\) and \(b+dz\) and so can be represented by a matrix \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\). It turns out that one can choose those elements so that the matrix is of the form \(\begin{bmatrix} ga & gk\\0& g\end{bmatrix}\) and this is the ideal that Lehman represents as \(g[a:k]\). See, for example, Propostition 5.2.1 in Henri Cohen’s  A Course in Computational Algebraic Number Theory.)
As this makes clear, Lehman is not averse to creating his own notations or his own nomenclature. For example, he talks about “quadratic rings” and “complete quadratic rings” where the more common terms are “order in a quadratic field” and “ring of integers of a quadratic field.” What I would call a fundamental discriminant is a “primitive discriminant.” His notation for a primitive quadratic form \(ax^2+bxy+cy^2\) is \((a:k)\), where \(k=b/2\) when \(b\) is even and \((b-1)/2\) when \(b\) is odd. The matrix corresponding to that quadratic form is \(\begin{bmatrix}2a & b\\b&2c\end{bmatrix}\), twice the usual choice. All of these choices are defensible, but they are also non-standard.
The notation \((a:k)\) for forms is chosen, of course, with the link between ideals and quadratic forms in view: the form \((a:k)\) and the ideal \([a:k]\) are related, but in a complicated way. Lehman’s account of the link is one of the clearest I have ever read, and much of the clarity arises exactly from the explicitness of the notation.
Another notable feature of this book is the sheer number of results that are left to the reader. Indeed, one can almost think of the book as a text for an inquiry-based course. Since so much of the material is described in terms of specific algorithms for computation, this makes sense: almost all results that boil down to a computation are left to the reader.
The test of a textbook, of course, is whether it can be taught. I am sure that Lehman’s approach can work well if the instructor buys into the approach fully. This is definitely not a book that allows one to pick and choose. Such a course is likely to be particularly successful with students who enjoy computing things by hand. Because this is one of the surprises in Lehman’s book: the emphasis on computation does not extend to using computers. There is no concern with whether the algorithms are actually practical in large cases, and there is no discussion at all of computer implementation. This might frustrate certain students: they are being taught how to compute by hand, but there is freely-available software that can do the same computations (but in a completely different notation) in seconds.
The nonstandard notation and nomenclature worry me a little. It would have been helpful to discuss alternative points of view in an appendix in order to prepare students for more advanced courses. In addition, while I am sure that Lehman is convinced that learning to compute by hand will help students to understand the material — and he may be right — I would have liked to see an appendix explaining how to do the same computations in GP and Sage.
As any reader who has followed me this far will notice, I am ambivalent about this book. In many ways, it is excellent, and I learned a lot by working through it. But I keep thinking that a similar approach using, say, the language and notations used by GP or Sage would work just as well pedagogically while preparing students for future work.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He has tried to teach algebraic number theory to undergraduates more than once, with wildly varying success.