Number Theory Seminar

Speaker: Peter Stevenhagen

Title: What is Rédei reciprocity?

Abstract: The quadratic reciprocity law, discovered by Euler and proved by Gauss, exhibits an unexpected symmetry in the arguments of a well-known quadratic symbol that carries the names of Legendre and Jacobi. In the 1930s, the Hungarian mathematician Rédei defined a quadratic symbol having three arguments, and applied it in the study of the quadratic class groups that have fascinated number theorists ever since they were defined in the Disquisitiones Arithmeticae.

In the 21st century, the Rédei symbol has been redefined in such a way that it becomes completely symmetric in its three arguments. The symbol has various applications, and plays a major role in the recent proof by Koymans and Pagano of my old conjecture on the solvability of the negative Pell equation.

I will explain the new definition and the resulting Rédei reciprocity law, and hint at the applications.