Number fields with prescribed norms

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.

arXiv comment41 pages, comments welcome! with an appendix by the last two authors

References in corpus (6)

• On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
• Nonabelian Cohen-Lenstra Moments
• A positive proportion of plane cubics fail the Hasse principle
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Cited by in corpus (1)

• Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensions