In his most recent book, Prof. Edwards introduces the readers to the content of the first few sections of Gauss’s Disquisitiones Arithmeticae; the basic topics covered are congruences, the Euclidean algorithm, prime decomposition, Euler’s phi function, and quadratic reciprocity. The presentation is algorithmic, not in the sense of, say, [A. Faisant, L’équation diophantienne du second degré. Paris: Hermann (1991; Zbl 0757.11012)], let alone [J. Buchmann, U. Vollmer, Binary Quadratic Forms. An algorithmic approach. Berlin: Springer (2007; Zbl 1125.11028)] or [H. Cohen, A course in computational algebraic number theory. Berlin: Springer (1993; Zbl 0786.11071)], but in the sense of Kronecker: everything defined should be computable exactly (that is, in terms of natural numbers) in a finite number of steps.
The main part of the book is about the theory of modules of “hypernumbers” (see also the author’s books [Fermat’s Last Theorem. Berlin: Springer (1996; Zbl 0904.11001) and Essays in Constructive Mathematics. New York, NY: Springer (2005; Zbl 1090.11001)], as well as his article “Composition of binary quadratic forms and the foundations of mathematics” in [C. Goldstein et al. (eds.), The shaping of arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae. Berlin: Springer (2007; Zbl 1149.01001)]. Hypernumbers are symbols of the form \(x + y \sqrt{A}\) in which \(x, y\), and \(A\) are nonnegative numbers. The basic objects are modules of hypernumbers, which are by definition finite lists of hypernumbers. Properties of modules are defined and proved in terms of such lists of generators; the main results obtained are a proof of the quadratic reciprocity law in Euler’s form: \((A/p) = (A/q)\) for primes \(p \equiv q \bmod 4A\), and the fact that binary quadratic forms of positive discriminant can be composed.
It is already clear from this short description that Edwards’ book is unusual in many ways. The most obvious difference between this book and a standard text is the author’s habit of inventing new names for familiar objects. Elements of real quadratic number fields are called hyperintegers, those elements whose coefficients with respect to the basis \(\{1, \sqrt{A}\}\) are nonnegative integers are called hypernumbers, genus characters are called signatures, instead of reduced or ambiguous modules the author introduces stable and pivotal modules, the Legendre symbol \((\frac Ap)\) is written in the form \(C_p(A)\), and the expression \(B^2 - AC\), which Gauss called the determinant of the form \(AX^2 + 2BXY + CY^2\) (like Gauss, Edwards assumes that the middle coefficient of his forms is an even number), is called discriminant of the form.
Another unusual aspect of the book is the author’s effort of avoiding negative numbers because they “have always led to metaphysical conundrums – why should a negative times a negative be a positive?” In fact, he writes, “I found that not only could I avoid negative numbers but that I didn’t miss them”. Of course, the basics of number theory as we can find them in Euclid, for example, can be presented without using negative numbers; and indeed, results like the fundamental theorem of arithmetic can be stated more simply within the natural numbers than within the ring of integers. But for an introduction to Gauss’s theory of binary quadratic forms based on modules, this approach is bound to fail: for positive definite forms like \(x^2 + y^2\), one needs modules in the ring \(\mathbb Z[\sqrt{-1}\,]\); indefinite forms like \(x^2 - 2y^2\), on the other hand, have the habit of possessing negative coefficients; avoiding this problem by replacing the form by an equivalent form with positive coefficients (see the footnote on p. 158) brings in negative numbers as coefficients of unimodular matrices.
The philosophical basis of the present book is Kronecker’s dream of arithmetization: everything in mathematics should ultimately be reduced to statements about natural numbers. To give an example, Edwards calls two hypernumbers \(a, b\) congruent modulo a module \(M\) if there exist \(N\)-linear combinations \(m, n\) of the hypernumbers in the list \(M\) such that \(a+m = b+n\) (note that a condition like \(a-b \in M\) does not make any sense: first of all, \(b = t+u\sqrt{A}\) can only be subtracted from \(a = r+s\sqrt{A}\) if \(r \geq t\) and \(s \geq u\), and \(M\) is a finite list of hypernumbers, not a module in Dedekind’s sense). Two such modules are called equal if they determine the same congruence relations.
Modern readers tend to identify Edwards’ modules with \(\mathbb Z\)-linear combinations of finitely many hypernumbers; Edwards avoids such infinite objects and expresses properties of modules in terms of a finite list of generators. All such properties defined in terms of generators must then be checked to be well defined, and conceptual proofs à la Dedekind become impossible. Instead, the author gives proofs by “arithmetical computation, [which], after all, is the purest form of deductive argument”.
What Edwards presents in this book is a version of the elementary aspects of Gauss’s Disquisitiones written by someone taking Kronecker’s dream seriously. For a more modern account, the reader may consult Dedekind’s treatment in Dirichlet’s Lectures on number theory [Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0936.11004)], or Bhargava’s elegant presentation of Gauss composition in [M. Bhargava, Lect. Notes Comput. Sci. 2369, 1–8 (2002; Zbl 1058.11030)].