The proportion of failures of the Hasse norm principle

TL;DR

This paper calculates the exact proportion of rational numbers that are locally norms everywhere but fail to be global norms from a given number field, providing precise quantitative insights into the Hasse norm principle failure.

Contribution

It provides the first explicit calculation of the exact proportion of such failures for any number field, advancing understanding of the Hasse norm principle.

Findings

Exact proportions are computed for various number fields.

Results reveal the frequency of local-global norm failures.

Quantitative data enhances the theoretical understanding of the Hasse norm principle.

Abstract

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.