Book review

The algebraic theory of numbers is constituted by attractive and instructive topics, which make the choice of chapters treated in a book on this subject very difficult. A new book on number theory gives a new point of view; the authors give an account on cubic and biquadratic fields (not only on quadratic fields), Gaussian periods, Brauer relations, module theory over a Dedekind domain, an algebraic number theory treatment of binary quadratic forms, tame ramification and the two-classgroup of a quadratic field.

The first four chapters and the final chapter (L-functions) have a theoretical nature, although the authors fix abstract ideas by means of worked examples (this is the new point of view). The remaining chapters are devoted to giving a detailed study of various arithmetic objects in situations of particular interest.

The history of the topics has influence on the development of the monography, although the authors whould like to make schematic expositions of the subject. For example, Diophantine equations now become an application of the theory; whereas, of course, historically they were the principal motivation for the development of the theory.

Some familiarity with elementary point set topology, with elementary Galois theory, and with basic module theory, including tensor products and structure theorem for finitely generated modules over principal domains are supposed from the part of the reader. In an undiluteded form, the book is best suited to a two-semester course at Masters level, but it may be used for a one-semester course, omitting much of the material and concentrating on the main theorems.

Chapter 1 is devoted to algebraic foundations (fields and algebras and the integrality and Noetherian properties), while Chapter 2 (Dedekind domains) gives some essential facts on valuations, completions and module theory over a Dedekind domain, containing the genesis of a whole host of ideas which are basic for the subject. An overview of the aims of algebraic number theory and the description of the motivation for the study it are obtained after reading these two chapters. One can see that many of the definitions and results of ordinary number theory have natural extensions in algebraic number theory and in fact are often better understood in this wider context (Dedekind domains). Frequently, the study of a suitable ring of algebraic numbers will help in solution of a problem which initially had been stated entirely in terms of ordinary integers: for instance, questions concerning the integral (or rational) coefficients can frequently be dealt with by the study of a suitable ring of algebraic integers. The algebraic number theory begins (in Chapter 2) by proving that, if L is a finite separable extension of the field of fractions K of a Dedekind domain d, then D, the integral closure of d in L, is a Dedekind domain. Applying this to Z, we can deduce that the ring of integers in an algebraic number field K is a Dedekind domain.

The behaviour of many concepts introduced in the previous chapter when they are extended from a given number field to an extended field is considered in Chapter 3, where the minimalist can omit much, but has to take into consideration Dedekind's theorem of the ramification of divisors of the discriminant and Kummer's criterion for the decomposition of prime ideals in an extension, as well as cyclotomic extensions. In this place, the central instance is the detailed analysis of methods of factorizing of the prime ideals. This problem is intimately related to the study of the arithmetic of finite extensions of fields when they are complete with respect to a discrete absolute value.

Chapter 4 is concerning the application of convex body theory to the study of classgroups and units (Minkowski methods) and is an usual chapter for such a book. Two results (which are basic in algebraic number theory and are useful for the subsequent chapters) are studied in detail in this chapter: a practical procedure for calculating classgroups and the Z-module structure of the unit group.

Chapter 5 is a complement to the chapter 4, gaining explicit informations on both fundamental units and classgroups for number fields of small degree: it is derived a detailed and intimate knowledge of the arithmetic of quadratic fields, biquadratic fields, cubic and sextic fields. Partially the techniques introduced here are used in the same chapter and in subsequent chapters for obtaining results on ordinary integered, but their full potential is realized when combined with the L-function results of chapter 7.

One of the most elegant chapters of the book is the chapter 6, devoted to the theory of cyclotomic fields, which play a fundamental role in a number of arithmetic problems: for instance primers in arithmetic progressions and Fermat's last theorem, throwing new light on the theory of quadratic fields and in particular providing the background to the "most natural" proof of the quadratic reciprocity law.

In chapter 7 various kinds of Diophantine equations as Fermat's last theorem, quadratic forms and cubic equations are considered. The last chapter is devoted to studying L-functions and Dedekind zeta-functions, being the most powerful methods in the whole book. By providing the history of attempts at solving the Fermat's last theorem, the book gives an excellent insight into the wealth of methods available in number theory. The Diophantine equations considered here and the methods used for solving them are instances of the very deep and beautiful arithmetic theory of elliptic curves.

Additional aids to the reader are the Appendix on character theory of Abelian groups and a wide range of exercises.

Written by well-known specialists in algebra and number theory, this book cannot be absent in any respectable library.