Colloquium Mathematics - Ted Chinburg, University of Pennsylvania

Title: The discriminants of number fields

Abstract:

The discriminant d_F of a number field F is a basic invariant of F. The smaller the absolute value |d_F| of d_F is, the more elements there are in the ring of integers O_F of size less than a given bound. This is relevant, for example, to cryptographic methods that use elements of O_F. I will discuss some of the history of lower bounds on |d_F| in terms of the degree n(F) of F. Toward the end of the talk I will discuss recent work with F. Bleher on efficient constructions of infinite families of F for which |d_F| grows slowly with n_F . I will also discuss work with X. Ding on the successive minima of the integers O_F of such fields F as well as on the non-randomness of the directions of these successive minima.