Constructing abelian extensions with prescribed norms

Christopher Frei, Rodolphe Richard

Research output: Contribution to journalArticlepeer-review


Given a number field K, a finite abelian group G and finitely many elements α1,…,αt ∈ K, we construct abelian extensions L/K with Galois group G that realise all of the elements α1,…,αt as norms of elements in L. In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate’s characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples

Original languageEnglish
Pages (from-to)381-399
Number of pages19
JournalMathematics of Computation
Issue number333
Publication statusPublished - 2021


  • Abelian extensions
  • Class field theory
  • Inverse problem
  • Norms form equations
  • Number fields

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Algebra and Number Theory


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