Chapter 7 - Rédei's Theorem

Summary

As a young mathematician, G. Hajós prepared a Ph.D. thesis on certain determinant identities. The chairman of his doctoral committee, L. Fejér, whose name is closely associated with Fourier analysis, feeling that the result did not match the outstanding talent of the candidate, rejected the thesis. This is why Hajós turned to Minkowski's famous unsolved conjecture.

In 1938 Hajós formulated the problem in terms of factorizations of groups and, making use of this reformulation, refuted Furtwängler's conjecture about multiple cube tilings, described in Chapter 1. This time his thesis met Fejér's legendary high standards.

Almost everyone, on first meeting the group theoretical equivalent of Minkowski's conjecture, tends to think that the solution of the problem should be immediate. So did Hajós. However, it took him three years to settle the conjecture. Looking back years later, he said that the problem had been extremely deceiving. It had offered many ways of attack but all but one led nowhere. Thinking about the problem almost constantly, he was able to pose it in many different versions. As he said, “When I had to walk up to the 5th floor I might make up my mind to find a new version on the way.”

Eventually he succeeded, obtaining his beautiful proof in 1941. It is so algebraic that there is no discernible connection between its lemmas and the geometry of the conjecture.