Number fields with prescribed norms

Christopher Frei (1) , Daniel Loughran (1) , Rachel Newton (2) , Yonatan Harpaz (3, 4) , Olivier Wittenberg (3, 5)

1 School of Mathematics - Department of Mathematics [Manchester]
2 Department of Mathematics and Statistics, University of Reading, RG66AX, Reading UK
3 CNRS - Centre National de la Recherche Scientifique
4 LAGA - Laboratoire Analyse, Géométrie et Applications
5 LMO - Laboratoire de Mathématiques d'Orsay

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.

Domaines: Théorie des nombres [math.NT]

Christopher Frei, Daniel Loughran, Rachel Newton, Yonatan Harpaz, Olivier Wittenberg. Number fields with prescribed norms. Commentarii Mathematici Helvetici, 2022, 97 (1), pp.133-181. ⟨hal-02373439⟩