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arXiv · 1810.06024

Number fields with prescribed norms

Christopher Frei·Daniel Loughran·Rachel Newton·Yonatan Harpaz·Olivier Wittenberg
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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.

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Christopher Frei, Daniel Loughran, Rachel Newton, Yonatan Harpaz, Olivier Wittenberg. 2018-10-14. Number fields with prescribed norms. https://doi.org/10.4171/cmh%2F528

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