Introduction to the English Edition of Hilbert’s Zahlbericht

David Hilbert’s (1862–1943) so-called Zahlbericht of 1897, which appears here for the first time in English, was the principal textbook on algebraic number theory for a period of at least thirty years after its appearance. Emil Artin, Helmut Hasse, Erich Hecke, Hermann Weyl and many others learned their number theory from this book. Even beyond this immediate impact Hilbert’s Zahlbericht has served as a model for many standard textbooks on algebraic number theory through the present day—cf. for instance Samuel’s little book, or the first chapter of Neukirch’s textbook. As a matter of fact, except for minor details (see Section 4 below), at least the first two parts of Hilbert’s text can still today pass for an excellent introduction to classical algebraic number theory. But even though Hilbert’s presentation definitely left its mark on this material, the text does remain a Bericht , i.e., a report, commissioned as it was by the Deutsche Mathematiker-Vereinigung (D.M.V., the German Mathematical Society), on the state of the theory around 1895. During the first years of its existence, the commissioning of comprehensive reports on all parts of mathematics was an important part of the activities of the D.M.V. The first ten volumes of the Jahresbericht contain thirteen such reports. David Hilbert and Hermann Minkowski were asked to write a joint report covering all of number theory. Eventually Hilbert and Minkowski decided to split the report, and it was agreed that Minkowski’s part should cover the elementary aspects of number theory like continued fractions, quadratic forms and the geometry of numbers. But in the end,