It is a fortunate fact that there is no shortage of textbooks on number theory, and when one has the opportunity to teach an undergraduate course on the subject there are plenty of excellent books to choose from. Depending on the approach, one may choose a textbook with many problems and exercises for the students, or maybe a new text with applications that attract newcomers to our beloved subject, or, why not, a classical text that brims with elegance in the choice of topics or proofs that will leave our students avid for more. The book under review falls on this last category, but also has plenty of well-chosen exercises at the end of every chapter.
This is a book written with love for the subject and with the presence of its readers (students) in mind all the time. The first five chapters treat the elementary arithmetic of the integers, starting with a chapter on the historical roots of the higher arithmetic and introducing some problems as motivation: The distribution of primes, Fermat’s Last Theorem and the theory of partitions. Thus, Grosswald goes from divisibility to the unique factorization theorem for the ring of integers, congruences modulo a fixed integer and the ring structure of the set of residue classes, Fermat’s and Euler’s theorems, and polynomial congruences on one variable culminating with Gauss’s law of quadratic reciprocity. Each chapter has a set of exercises and a bibliography pointing to other approaches, including textbooks, classical works or more advanced books. Whenever possible the author gives alternative proofs. For example, when discussing arithmetic functions, Dirichlet’s series are used to prove a property about the summation of the Mobius function for the divisors of a given integer.
The last part of the book is devoted to some topics from analytic and algebraic number theory. The prerequisites for the student now include some elementary real and complex analysis and an acquaintance with algebra. The rewards are a beautiful discussion of Riemann’s zeta function, including Euler’s product formula, the functional equation and the analytic continuation of the zeta function to the whole complex plane (with the exception of a simple pole at s = 1), its relation to the distribution of primes. Finally, after some additional preliminaries, e.g., a simple Tauberian theorem, one reaches the highlights: the statement and proof of the prime number theorem and the proof of Dirichlet’s theorem on primes in arithmetic progressions.
On the algebraic side, assuming elementary properties of commutative rings and fields, number fields and their rings of integers are introduced. Examples of quadratic and cyclotomic extensions are discussed with some detail, proceeding then to prove the uniqueness of the factorization of ideals of a ring of integers into prime ideals and the finiteness of the ideal class group. A highlight of this approach is the proof of Fermat’s Last Theorem for regular primes.
The chapter on Diophantine equations illustrates a beautiful trend on number theory: The spectacular developments since the 1984 second edition of the book include now what was breaking news in a footnote on page 262: Mordell’s conjecture is now Faltings’ theorem. Catalan’s conjecture is now Mihailescu theorem and Fermat’s last conjecture is now Wiles’ theorem. There was, however, time to include the results of Martin Davis, Julia Robinson and Yuri Matijasievic on the non-existence of a general algorithm to decide if a given Diophantine equation has or does not have solutions.
This chapter is mix of elementary stated and proved results and an introduction to more advanced topics such as the arithmetic of curves over number or finite fields. This is developed to the point of being able to formulate, for example, the Mordell-Weil theorem that under the “chord and tangent” operation the set of rational points on a curve of genus 1 is a finitely generated abelian group, and to discuss some aspects of Weil conjectures and theorems (for the case of curves).
This is a reprint of the 1984 second edition of a book whose first edition was published in 1966 (MacMillan). The major differences between the second and first editions are the addition of three totally new chapters (10, 11 and 13), the merging of the old chapters 9 and 11 into the new chapter 14, and some parts of the old appendices into the new chapter 10.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org