There are many different ways to do history of mathematics… and, to paraphrase Kipling, every single one of them is right! The different approaches simply ask different questions and look at different pieces of evidence. Among historians, the current fashion is to emphasize cultural context, analyze and discuss the sociology of knowledge, and to try to capture how the development looked like at the time, without putting too much reliance on hindsight. Mathematicians, on the other hand, tend to prefer a "history of ideas" approach in which one follows the development of the actual mathematical ideas and tries to figure out how we got where we are.
The book under review is an example of how the latter approach, done well, can offer a real contribution. The history of ideas approach is particularly fruitful when it is highly focused; in this case, the focus is on one specific theorem and the four persons involved in proving it. The theorem says that every finite-dimensional central simple algebra over a number field is cyclic. The persons are Richard Brauer, Emmy Noether, Helmut Hasse, and, a surprise appearance, Adrian Albert.
The Brauer-Hasse-Noether theorem, as it has come to be known, is one of the few for which we have a precise birth date: all the evidence points to November 9, 1931 as the day in which it was proved. Two factors make it possible to know this: first, the collaboration between Albert, Brauer, Hasse, and Noether was mostly conducted by correspondence, much of which has survived; second, Hasse was eager to include the theorem in a special issue of Crelle in honor of his teacher and mentor Kurt Hensel. The second factor made for very rapid writing and publication of the paper (the deadline for submissions to the special issue had long passed!), and the first allows us to see the progress towards the proof.
Roquette's account is quite interesting, both mathematically and in terms of the personal interaction of the four mathematicians. In order to explain why the theorem was so exciting, he has to sketch the intellectual context, which has to do with the local-global principle, the birth of local class field theory, and Noether's interest in non-commutative algebras as tools for both number theory and representation theory. So one can learn quite a bit about the actual mathematics here, including the rather amazing story of the role of the Grunwald-Wang theorem. Roquette is very sensitive to the fact that steps that seem "clear" to someone used to "structural" algebra were not at all clear (except maybe to Noether) at the time. While he doesn't go into the questions about how techniques become part of the standard toolkit of mathematicians, he does offer some interesting evidence for historians interested in that question.
The personal dynamics are also interesting. In particular, Roquette demonstrates that Albert only got left out of the final paper because communication with him was so much slower than among the three Germans. While letters from Noether to Hasse tended to arrive the day after they were written, letters from Albert (and even American journals) took several weeks to show up. Given the rush to get a paper into the Hensel issue and the fact the three German mathematicians were as yet unaware that Albert had also solved the problem, the paper had only three names on it, and a footnote indicating Albert's contribution was added in proof. He offers less detail about the actual content of Albert's work, though there are some tantalizing hints that his approach was quite different from the Germans', being at once more general (Albert's theorems tended to be about general fields whenever possible) and less "structural" (Roquette comments that Albert did mathematics in the "Dickson style").
Roquette's book is very much a "local" contribution to the history of modern number theory, and in particular to the story of the impact of Emmy Noether's "structural" approach on the subject. In fact, it largely takes the latter story as a given, not probing too deeply into the question of how a rather unique point of view spread from Göttingen to the rest of the mathematical world and came to become the dominant point of view. Anyone who does want to understand that story, however, will find Roquette's book a crucial source of insight and information, and any mathematician who wants to understand why a number theorist would get excited about a theorem dealing with division algebras will find the issue well explained. I learned a lot by reading it, and I hope that Roquette will continue to explore this interesting historical period.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He has a strong interest in the history of algebra and number theory in the 20th century.