The archives of the University of Göttingen are a treasure-trove of documentary material for historians of mathematics, especially with regard to algebra and number theory in the early 20th century. From that archive, we have already had an edition of the correspondence between Helmut Hasse and Emmy Noether. Now Peter Roquette and Günther Frei have done the same for Hasse’s correspondence with Emil Artin between 1923 and 1934.

The correspondence reprinted here comes from Hasse’s papers, and as a result it contains only the letters that Artin wrote to Hasse. There is an irony here: of the two, Hasse was the more dedicated and more frequent letter-writer, while Artin clearly preferred to discuss ideas in person. Perhaps this is why Hasse kept all of Artin’s letters while those sent to Artin seem to have been lost.

The letters mostly concern class field theory: Artin’s discovery of his reciprocity law, Hasse’s use of that result to produce a general n-th power reciprocity law, the relation between local and global class field theory, and so on. Both Artin and Hasse were interested in other mathematical questions, but the letters dealt only with their common interests. After 1934, the correspondence broke off and was never resumed.

The letters are extensively annotated with commentary on the state of the questions and on Hasse’s mathematical activities at the time. For example, Artin’s letter of October 27, 1927 is one and a half pages and a four page enclosure; the commentary is nine pages long, and includes a photograph of a 1927 page from Hasse’s journal. (Shall we eventually see an edition of that as well? Let’s hope so!)

The first section of the book includes an introduction, brief biographical accounts of Artin and Hasse, articles about class field theory and reciprocity laws, and a timeline. Everything in the book is in German except for the introduction and the biography of Artin, which are in English. There are indices of names and keywords and an extensive bibliography.

Editions like this one are routine in the humanities, but quite rare in mathematics. There should be more. By reprinting the material in the archives and providing such extensive ancillary materials the editors have done a great service to everyone interested in the history of algebra and number theory.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.