## Constructing abelian extensions with prescribed norms

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Christopher Frei and Rodolphe Richard
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## Abstract:

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha _1,\ldots ,\alpha _t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha _1,\ldots ,\alpha _t$ 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.

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## Additional Information

**Christopher Frei**- Affiliation: TU Graz, Institute of Analysis and Number Theory, Steyrergasse 30/II, 8010 Graz, Austria
- MR Author ID: 938397
- ORCID: 0000-0001-8962-9240
- Email: frei@math.tugraz.at
**Rodolphe Richard**- Affiliation: 13, rue du Croisic, 22200 Plouisy, Bretagne, France
- Address at time of publication: University College London, 25 Gordon Street, WC1H 0AY London, United Kingdom
- MR Author ID: 875874
- Email: rodolphe.richard@normalesup.org
- Received by editor(s): July 2, 2020
- Received by editor(s) in revised form: December 14, 2020, and April 8, 2021
- Published electronically: July 22, 2021
- Additional Notes: The first author was supported by EPSRC grant EP/T01170X/1. The second author was supported by ERC grant GeTeMo 617129, and Leverhulme Research Project Grant “Diophantine problems related to Shimura varieties”
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 381-399 - MSC (2020): Primary 11Y40, 11R37; Secondary 14G05, 11D57
- DOI: https://doi.org/10.1090/mcom/3663
- MathSciNet review: 4350543