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**Elementary and analytic theory of algebraic numbers.
3rd ed.**
*(English)*
Zbl 1159.11039

Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-21902-1/hbk). xi, 708 p. (2004).

This wonderful textbook appeared in 1974 for the first time (see Zbl 0276.12002), the second edition in 1990 (see Zbl 0717.11045) and it is a very great pleasure to have the third edition at hand in 2004. This proves that the book has already been a ‘Bible’ for generations of number theorists. No matter if someone has his special interest in algebraic number theory, analytic number theory, Diophantine equations or whatever: this book has always turned out to be crucial. It contains in a very clear and general exposition basic knowledge about valuations (explained in general Dedekind domains), algebraic numbers, geometric number theory, units of number fields, ideal classes, extensions of number fields, the \(p\)-adic fields and its applications. Moreover it gives a detailed description of zeta functions, class number formula and the Siegel-Brauer theorem. The Appendix extends the text into some connected important directions. The detailed author index and subject index, list of notation makes it easy to navigate in the book if e.g. someone just wants to look up a special notion or assertion in number theory.

But Narkiewicz’s celebrated book is much more than a wonderful textbook. The most important values of the book come just after the text. There is a list of Problems (35 in the first edition, 49 in the second edition, 60 in the present edition) all of which represent a current research direction of number theory. The most delicate part of the book is certainly the list of References, which was substantially extended in all new editions: it ranges now from page 535 to page 683, in 8pt type! This list represents a selection of the most interesting articles and books in number theory. It is really an honour to be listed in the References of this book.

But Narkiewicz’s celebrated book is much more than a wonderful textbook. The most important values of the book come just after the text. There is a list of Problems (35 in the first edition, 49 in the second edition, 60 in the present edition) all of which represent a current research direction of number theory. The most delicate part of the book is certainly the list of References, which was substantially extended in all new editions: it ranges now from page 535 to page 683, in 8pt type! This list represents a selection of the most interesting articles and books in number theory. It is really an honour to be listed in the References of this book.

Reviewer: István Gaál (Debrecen)