The Story of Algebraic Numbers in the First Half of the 20th Century

The book under review is a companion to the author’s earlier Rational Number Theory in the 20th Century: From PNT to FLT. Both books are works of extraordinary scholarship: the earlier book had 6849 references while this one has 4482. In both books, each reference corresponds on average to about one-twentieth of a page of the main text. The pace is thus extremely brisk, with the general topic changing every page or two. In this way and others, the two volumes are reminiscent of Dickson’s classic three-volume History of the Theory of Numbers. However, the mathematics in Narkiewicz’s two books is much more modern, and thus much more sophisticated.

While algebraic numbers in the first half of the 20th century certainly form the focus of the book under review, considerable attention is given also to both earlier and later times. A long Chapter 1 sets the stage with a survey of algebraic number theory through roughly 1890. Chapter 2 starts with Hilbert’s very influential 1897 Zahlbericht and gives equal attention to Hensel’s slightly later introduction of \(p\)-adic numbers. Chapters 3–6 proceed methodically from 1900 to 1950, with major themes including the development of class field theory and associated \(L\)-functions. Tate’s 1950 thesis is one of the highlights of Chapter 6 and serves as an fitting endpoint; it centers on an approach to Hecke’s \(L\)-functions that treats each prime separately via \(p\)-adic numbers.

The entire text alternates between a larger and smaller font. Text in the larger font focuses on the principal developments and stays within the current chapter’s time frame. Text in the smaller font centers on subsequent developments, often taken up to the present day. Many readers will feel that they are brought closer to the main narrative by the recurring presence of direct connections to contemporary issues.

The book has a just-the-facts feel from start to end. With its enormous content and terse style, it is not a single story in any conventional sense. Rather it briefly summarizes hundreds of individual stories and indicates how these stories are interwoven. It points with great precision to fuller versions of these stories in the literature. For example, biography is not at all the subject of this book, with biographical information being limited to footnotes of a common form, represented by “Erich Hecke (1887–1947), professor in Göttingen and Hamburg. See [3266, 3662].”

As the title indicates, it is mathematics which is the subject of this book. Here the summaries of individual stories are detailed enough to make the general arc of each story clear, including sometimes plot twists. To give readers of this review a more exact sense of the book, the next three paragraphs present three of its stories concerning how the class number \(h_n\) of the \(n^{\rm th}\) cyclotomic field \(Q(\exp(2 \pi i/n))\) varies with \(n\). The first two stories actually concern the class number \(h_n^+\) of the real subfield \(Q(\cos(2 \pi/n))\). The last one concerns the complementary factor \(h_n^- = h_n/h_n^+\), which Kummer proved to be an integer. The summaries given in the book each contain considerably more information than their briefer versions given here.

The first story is told on page 132 about the conjecture \(h_{2^k}^+=1\) for all \(k\). Weber established this equality for the first nontrivial case \(k=4\). In one of his famous books from the 1890s, he conjectured that it was false for \(k=6\). Despite this actual history, the general conjecture is often attributed to Weber. It was proved by various authors starting in 1960 for \(k=5\), \(6\), and \(7\). It was more recently proved under the generalized Riemann hypothesis for \(k=8\). The state of the art now is that it is known to be true for \(k=8\) and has been proved conditionally for \(k=9\). In a sequence of three papers around 2010, Fukuda and Komatsu showed that there is at least some truth in the general conjecture, as any prime dividing an \(h_{2^k}^+\) has to be at least a billion.

The second story is told on page 179 and starts with work of Vandiver in 1929. From this work grew Vandiver’s conjecture that no prime \(p\) divides its corresponding \(h_p^+\). Seeing as the \(h_n^+\) are generally small, it was thought at one point that even the stronger statement \(h_p^+ < p\) might be true. However Schoof and Washington found that this inequality fails for \(p=641491\) in 1988. The actual Vandiver conjecture remains a possibility and it was verified by Buhler and Harvey for \(p\) less than 163 million in 2011. The pointers to the literature let readers rapidly find out more. For example, Schoof and Washington specialized a family of cyclic quintic polynomials due to Emma Lehmer to conclude that \(h_{641491}^+\) is divisible by \(1566401\).

The third story, like many in the book, is told in several episodes. Pages 23–25 report how Kummer around 1850 already had an explicit formula for \(h_p^-\) and conjectured its asymptotic equivalence with the rapidly growing function \(L(p) = 2 p \left( \frac{p}{ 4 \pi^2} \right)^{(p-1)/4}\). Pages 131–132 describe many developments including a particularly elegant 1913 formula of Maillet for \(h_p^-\). This very explicit formula would let an engaged reader easily see on a computer how \(h_p^-/L(p)\) has a strong tendency to be near \(1\). Pages 242–244 take readers into the modern era. Various results say that indeed \(L(p)\) approximates \(h_p^-\) quite closely in several rigorous ways. However a heuristic argument of Granville from 1990 says that the originally conjectured \(\lim_{p \rightarrow \infty} h_p^-/L(p)=1\) is quite likely to be false.

A clear theme of the book is that progress in algebraic numbers is hard-won but unmistakable as the years pass. Narkiewicz never injects personal opinions into his narrative, leaving individual readers to build their own viewpoints from the multiple stories the book summarizes. The back cover modestly says that this book “may be helpful in preventing rediscoveries of old results.” I’d say it has the much deeper purpose of increasing our community’s understanding of our common enterprise.

David P. Roberts is Professor of Mathematics at the University of Minnesota, Morris