Number theory: algebraic numbers and functions (Q2785516)

This is a textbook aimed at readers with good knowledge of linear algebra plus some supplementary studies in field theory roughly up to the level of Galois theory. The leading principles of the author are: First, the author wants to place emphasis on the history of the subject, in fact, he wants the reader to follow the historical development of number theory. Second, the case of an algebraic function field of one variable over a perfect constant field should be treated parallel with number fields. This is particularly useful in a modern elementary textbook in view of the growing interest in proving function field analogues of known theorems about number fields. Third, the author wants to provide the reader with all necessary basic tools required in research work. However, the edifice of class field theory is not included, only a short outline is given. NEWLINENEWLINENEWLINEChapter 1 contains a sample of theorems in old elementary number theory with two exceptions: public key cryptology is included in order to give a modern application, and the prime number theorem is proved in a simplified form originated by Newman and Zagier. NEWLINENEWLINENEWLINEChapter 2 deals with the part of the theory which is valid in different orders of a number field. This corresponds roughly with the state of the science before Dedekind. Minkowski's geometry of numbers and Dirichlet's unit theorem are included here. NEWLINENEWLINENEWLINEChapter 3 contains Dedekind's ideal theory formulated so that it allows a unified treatment of number fields and function fields. In Chapter 4 the author discusses basic valuation theory. NEWLINENEWLINENEWLINEIn Chapter 5 there is an exposition of the function field case. The given proof of the Riemann-Roch theorem is a particularly readable one; it is a simplification of the proof in \textit{H. Hasse}'s monumental ``Zahlentheorie'' [Akademie-Verlag (1949; Zbl 0035.02002)]. In the author's opinion, various well known textbooks, dealing with basic arithmetical function field theory are not suitable for a novice, and one cannot but agree. NEWLINENEWLINENEWLINEIn Chapter 6 the Dedekind-Hilbert theory of number fields and their finite normal extensions is completed by introducing the decomposition group, inertia group and ramification groups. This allows for a short treatment of cyclotomic fields. The Kronecker-Weber theorem is the content of five exercises. NEWLINENEWLINENEWLINEThe material in Chapter 7 is deeper. First, adeles and ideles are defined, and Grössencharacters are introduced following Hecke's original definition. The interpretation as idele class group characters is explained. The bulk of the chapter is devoted to Tate's thesis and to the functional equation for Hecke \(L\)-series. Functional equations in other cases are then obtained as special cases. This complies with the author's first and also the third principle to provide the important tools for the reader, but in the reviewer's opinion it is debatable whether a proof of the functional equation for the Dedekind zeta function, say, derived from Tate's thesis is suitable for a novice. In any case, this method gives the book a peculiar characteristic. It is of interest to make a comparison with \textit{J. Neukirch's} book ``Algebraische Zahlentheorie'' [Springer-Verlag (1992; Zbl 0747.11001)]. Neukirch refrains from an exposition of Tate's thesis and for didactic reasons proves the functional equation four times: for the Riemann and Dedekind zeta functions, and for the Dirichlet and Hecke \(L\)-series separately. NEWLINENEWLINENEWLINEChapter 8 deals with some applications of Hecke \(L\)-series, and Chapter 9 contains a rather detailed discussion of quadratic fields including continued fractions, Gaussian sums and class number formulas. In Chapter 10 there is a short overview of class field theory. NEWLINENEWLINENEWLINEThe book contains a large amount of material in a compact form, and the author's style is short and to the point. He has the rare but important ability of making complicated things look simple. Each chapter ends with a bunch of exercises. One thing worth criticizing is the rather short index, which should be much expanded. It is difficult to find the definitions of some concepts.