Traditionally, the words “collaboration,” and “principal theorem in division algebras in the 1930s” are associated with the celebrated German trio of mathematicians, Richard Brauer, Helmut Hasse and Emmy Noether. In fact, Brauer, Hasse, and Noether formed one of the collaborative efforts that led to the proof of the principal theorem in linear algebras in the 1930s, that is, the structural description of normal division algebras over an algebraic number field. This paper, however, highlights the other joint work linked with the proof of this theorem, namely that of A. Adrian Albert and Hasse. The paper’s title alludes to a tension in this collaboration. It also refers to the delicate rapport Hasse and Albert cultivated in their correspondence. The seemingly cordial friendship they projected on paper helped keep potentially difficult issues in balance and, consequently, preserved their mutual exchanges. Our specific avenue of investigation emphasizes the correspondence from Albert to Hasse in 1931 and early 1932. At that time, Brauer, Hasse and Noether worked to prove the principal theorem in Germany. Ultimately, in late 1931, this triumvirate established the principal theorem that every normal division algebra over an algebraic number field of finite degree is cyclic [Brauer et al. 1932]. Two months later, Albert and Hasse published a joint work with an alternative proof, entitled “A Determination of all Normal Division Algebras Over an Algebraic Number Field” which also included something of a brief history of the theorem [Albert and Hasse 1932].
In the current paper, we use new archival materials to begin to unravel the behind-the-scenes history of this truly international development in the history of mathematics. In the first two sections of this paper we introduce the protagonists of our research: Albert and Hasse. We trace the individual mathematical paths Albert and Hasse each traveled to arrive at the central question in division algebras. In particular, we see the strong influence of Leonard Eugene Dickson and Joseph Henry Maclagan Wedderburn on the mathematical work of Albert and the prominent role of Kurt Hensel’s \( p \)-adic number theory in Hasse’s mathematical researches. A more formal discussion of the mathematical concepts related to the theory of normal simple algebras in the late 1920s and early 1930s forms the focus of the following section. With this framework in place, we turn our attention to the actual correspondence from Albert to Hasse. These letters contribute to a clearer understanding of the development of a proof of the “principal theorem” in the theory of linear algebras in those waning months of 1931 in Germany and Chicago. Finally, we offer some reflective, concluding remarks on this history.
