Decades ago by now, I had the great good fortune of learning the nuts and bolts of algebraic number theory (without class-fields) from Erich Hecke’s Vorlesungen über die Theorie der Algebraischen Zahlen, about which André Weil said, in the Foreword to his Basic Number Theory, “To improve upon Hecke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task.” I intend to place my copy of Wladyslav Narciewicz’ beautiful book, Elementary and Analytic Theory of Algebraic Numbers, next to Hecke’s on my shelves: Narkiewicz hits the same mark in our day that Hecke hit some eighty years ago.
Narkiewicz’ tome, weighing in at xi + 708 pages, is an imposing work, ambitious in scope and even encyclopaedic (at its intended level), despite the author’s emphatic and perhaps anachronistic avoidance of class-field theory. I think he makes the right choice, however, given that probably the most popular route to class-field theory involves group cohomology (if only just a smidgen, as in e.g. Michiel Hazewinkel’s elegant treatment). Avoiding cohomology in setting up class-field theory is of course not impossible: in fact, see Weil’s Basic Number Theory, for instance, or Hasse’s treatment in his (also very thick) book, Number Theory. However, to do so is arguably perhaps an unnatural act — or at least needlessly demanding of the relative beginner. It is reasonable to imagine algebraic number theory as partitioned into two parts, class-field theory and its (huge) complementary set, which, both on technical and pedagogical grounds, is fairer game. This split is natural also on the count that class-field theory’s frontier consists largely in the persistently daunting problem of developing the non-abelian case and therefore it is probably prudent for a fledgling arithmetician (or even an old-timer) to avoid this area, or at least postpone it until after an extended pilgrimage through the territory covered by Narkiewicz has been completed.
To be sure, the material Narkiewicz deals with is so beautiful and deep that the aforementioned arithmetician should more than likely be inclined to stay and play for a while. And, to make matters worse, Narkiewicz sweetens the pot by appending, on pp. 529–534, a list of <60 open problems (in his list of 60, a positive number of problems are presented as having been solved between editions of the book), followed by a titanic bibliography of almost thirty pages (pp. 535-683); the coverage takes in both classical and modern themes: there are four entries for Gauss, C. F., and three for Wiles, A. The book succeeds spectacularly in making clear that the favored part of algebraic number theory not only has unparalleled historical roots but supports vast and beautiful current research.
When it comes to the material’s specific presentation, Elementary and Analytic Theory of Algebraic Numbers is also well-written and eminently readable by a good and diligent graduate student. It would serve beautifully for a graduate-level course in number theory sans class-field theory, with the obvious caveat that a 700-page book is too long to permit a complete word-for-word treatment in a semester (or even a year?). But Narkiewicz’ presentation is so clear and detailed that coverage of certain topics, if only in part, is extremely beneficial. The hungrier reader is then certainly afforded the opportunity to go further into a subject of choice, both in the book’s pages themselves and via the references.
And the choices of subjects are delectable, as indicated by the book’s chapters: Dedekind domains and valuations, algebraic numbers and integers, units and ideal classes, extensions, local fields (including a section on harmonic analysis!), applications of local fields (including material on adèles and idèles), analytic methods (including Chebotarev’s Theorem), abelian fields (including sections titled, “The Class-number Formula and the Siegel-Brauer Theorem” and “Class-numbers of Quadratic Fields”), and factorizations. Each chapter ends with a section of “Notes” each of which is an exemplar of historical scholarship consonant with Narkiewicz’ erudition in these disciplines. These Notes provide a wonderful road-map through the literature, again underscoring the incomparable value of this book as both an unparalleled introduction to and spring-board for research on algebraic number fields void of class-field theory.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.