Number fields with prescribed norms: with an appendix by Y. Harpaz and O. Wittenberg

Christopher Frei, Daniel Loughran*, Rachel Newton

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)138-181
Number of pages44
JournalCommentarii Mathematici Helvetici
Issue number1
Publication statusPublished - 2022


  • class field theory
  • harmonic analysis
  • Hasse norm principle
  • rational points on varieties

ASJC Scopus subject areas

  • Mathematics(all)

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