Number fields with prescribed norms

TL;DR

This paper investigates the distribution and existence of number field extensions with a fixed abelian Galois group where certain elements are norms, demonstrating the Hasse norm principle holds for all such extensions when ordered by conductor.

Contribution

It establishes the existence of these extensions and proves the Hasse norm principle holds for all G-extensions of a number field ordered by conductor.

Findings

Hasse norm principle holds for 100% of G-extensions ordered by conductor.

Existence of extensions with prescribed norm conditions is proven.

Provides a geometric proof of the existence result.

Abstract

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.