André Macedo

Explicit methods for the Hasse norm principle and applications

Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions.

In this talk, I will present recent work developing explicit methods to study this principle for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will also discuss the geometric interpretation of this principle and how it relates to the weak approximation property for norm one tori.

This is joint work with Rachel Newton.